SKEW CALABI-YAU ALGEBRAS AND HOMOLOGICAL IDENTITIESmreyes/smashproduct.pdfSKEW CALABI-YAU ALGEBRAS AND HOMOLOGICAL IDENTITIES 3 where l 2Zwand 1A (l) is the shift of 1A by degree l,

SKEW CALABI-YAU ALGEBRAS AND HOMOLOGICAL

IDENTITIES

MANUEL REYES, DANIEL ROGALSKI, AND JAMES J. ZHANG

Abstract. A skew Calabi-Yau algebra is a generalization of a Calabi-Yau al-

gebra which allows for a non-trivial Nakayama automorphism. We prove threehomological identities about the Nakayama automorphism and give several ap-

plications. The identities we prove show (i) how the Nakayama automorphism

of a smash product algebra A#H is related to the Nakayama automorphismsof a graded skew Calabi-Yau algebra A and a finite-dimensional Hopf algebra

H that acts on it; (ii) how the Nakayama automorphism of a graded twist of A

is related to the Nakayama automorphism of A; and (iii) that the Nakayamaautomorphisms of a skew Calabi-Yau algebra A has trivial homological deter-

minant in case A is noetherian, connected graded, and Koszul.

0. Introduction

While the Calabi-Yau property originated in geometry, it now has incarnationsin the realm of algebra that seem to be of growing importance. Calabi-Yau tri-angulated categories were introduced by Kontsevich [Ko] in 1998. See [Ke] for anintroductory survey about Calabi-Yau triangulated categories. Calabi-Yau alge-bras were introduced by Ginzburg [Gi] in 2006 as a noncommutative version ofcoordinate rings of Calabi-Yau varieties. Since the late 1990s, the study of Calabi-Yau categories and algebras has been related to a large number of other topicssuch as quivers with superpotentials, DG algebras, cluster algebras and categories,string theory and conformal field theory, noncommutative crepant resolutions, andDonaldson-Thomas invariants. Some fundamental questions in the area were an-swered by Van den Bergh recently in [VdB3].

One known method for constructing a noncommutative Calabi-Yau algebra is toform the smash product A#H of a Calabi-Yau algebra A with a Calabi-Yau Hopfalgebra H that acts nicely on A. This phenomenon has been studied quite broadly;for instance, see [BSW], [Fa, Section 3], [IR, Section 3], [LM], [LiWZ], [WZhu], and[YuZ1, YuZ2]. One of the most general results in this direction [LiWZ] states thatunder technical hypotheses on Calabi-Yau algebras A and H, A#H is Calabi-Yauif and only if the homological determinant of the H-action on A is trivial. However,the smash product A#H may be Calabi-Yau even when A is not Calabi-Yau; see,for instance, Example 1.6 below. In order to explain this phenomenon, it is naturalto look into a broader class of interesting algebras, called skew Calabi-Yau algebras

Date: February 8, 2013.2000 Mathematics Subject Classification. Primary 16E65, 18G20, Secondary 16E40, 16W50,

16S35, 16S38.Key words and phrases. Calabi-Yau algebra, Skew Calabi-Yau algebra, Artin-Schelter regu-

lar, Artin-Schelter Gorenstein, Nakayama automorphism, homological determinant, homologicalidentity.

1

2 MANUEL REYES, DANIEL ROGALSKI, AND JAMES J. ZHANG

in this paper, which are a generalization of Ginzburg’s Calabi-Yau algebras. WhileGinzburg originally defined the Calabi-Yau property for DG-algebras [Gi, Definition3.2.3], we only consider the non-DG case in this paper.

We will employ the following notation. Let A be an algebra over a fixed com-mutative base field k. The unmarked tensor ⊗ always means ⊗k. Let M be anA-bimodule, and let µ, ν be algebra automorphisms of A. Then µMν denotes theinduced A-bimodule such that µMν = M as a k-space, and where

a~m~ b = µ(a)mν(b)

for all a, b ∈ A and all m ∈ µMν(= M). Let Ae denote the enveloping algebraA ⊗ Aop, where Aop is the opposite ring of A. An A-bimodule can be identifiedwith a left Ae-module naturally, or with a right Ae-module since Ae is isomorphic to(Ae)op. We usually work with left modules, unless otherwise stated. For example,

HomA(−,−) and ExtdA(−,−) are defined for left A-modules. A right A-module isidentified with a left Aop-module. Let B-Mod be the category of left B-modulesfor a ring B.

Definition 0.1. Let A be an algebra over k.

(a) A is called skew Calabi-Yau (or skew CY, for short) if(i) A is homologically smooth, that is, A has a projective resolution in the

category Ae-Mod that has finite length and such that each term in theprojective resolution is finitely generated, and

(ii) there is an integer d and an algebra automorphism µ of A such that

(E0.1.1) ExtiAe(A,Ae) ∼=

{0 i 6= d1Aµ i = d,

as A-bimodules, where 1 denotes the identity map of A.(b) If (E0.1.1) holds for some algebra automorphism µ of A, then µ is called

the Nakayama automorphism of A, and is usually denoted by µA. It is nothard to see that µA (if it exists) is unique up to inner automorphisms of A(see Lemma 1.7 below).

(c) [Gi, Definition 3.2.3] We call A Calabi-Yau (or CY, for short) if A is skewCalabi-Yau and µA is inner (or equivalently, µA can be chosen to be theidentity map after changing the generator of the bimodule 1Aµ).

There is some variation in the literature concerning the exact definition of (skew)CY algebras. Ginzburg included in his definition of a CY algebra [Gi, Definition3.2.3] the condition that the Ae-projective resolution of A is self-dual, but Van denBergh has shown that this is satisfied automatically [VdB3, Appendix C]. We arealso not the first to study skew CY algebras; the notion has been called twisted CYin several recent papers (see the beginning of Section 1 for more discussion).

We are primarily interested here in the special case of graded algebras A. If Ais Zw-graded, Definition 0.1 should be made in the category of Zw-graded modulesand (E0.1.1) should be replaced by

(E1.0.1) ExtiAe(A,Ae) ∼=

{0 i 6= d,1Aµ(l) i = d,

SKEW CALABI-YAU ALGEBRAS AND HOMOLOGICAL IDENTITIES 3

where l ∈ Zw and 1Aµ(l) is the shift of 1Aµ by degree l, and here l is called the ArtinSchelter (AS) index. In this case, the Nakayama automorphism µ is a Zw-gradedalgebra automorphism of A.

For simplicity of exposition, in the remainder of this introduction A is a noe-therian, connected (that is, A0 = k) N-graded skew CY algebra. Equivalently (aswe will note in Lemma 1.2), A is a noetherian Artin-Schelter regular algebra. Inthis case µA is unique since there are no non-trivial inner automorphisms of A.Suppose further that A is a left H-module algebra for some Hopf algebra H, whereeach graded piece Ai is a left H-module. Let A#H be the corresponding smashproduct algebra; we review the definition in Section 2, but we note here that itis the same as the skew group algebra A o G in the important special case thatH = kG is the group algebra of a group G acting by automorphisms on A. There isalso a well-established theory of homological determinant of a Hopf algebra action[JoZ, JiZ, KKZ] that determines a map hdet : H → k; this is reviewed in Section 3.

Our first result provides an identity for the Nakayama automorphism of A#Hin terms of those of A and H, along with the homological determinant hdet of theaction. It helps to explain how the smash product A#H may become Calabi-Yaueven when A is only skew Calabi-Yau. The idea is that even if µA is not an innerautomorphism, µA#H may become inner; see also Corollary 0.6 below.

Theorem 0.2. Let H be a finite dimensional Hopf algebra acting on a noetherianconnected graded skew CY algebra A, such that each Ai is a left H-module and Ais a left H-module algebra. Then

(HI1) µA#H = µA#(µH ◦ Ξlhdet),

where hdet is the homological determinant of the H-action on A, and Ξlhdet is thecorresponding left winding automorphism (see Section 1).

Theorem 0.2 will be proved in Section 4. In the special case where µA = IdAand µH = IdH , Theorem 0.2 partially recovers some results in a number of papers[LM, Fa, WZhu, LiWZ]. A natural question is if Theorem 0.2 holds when H isinfinite dimensional or when A is ungraded. When H is an involutory CY Hopfalgebra and A is an N -Koszul CY algebra, the question was answered in [LiWZ,Theorem 2.12], but in general the question is open.

By the term homological identity which we use in the title of the paper, we meanan equation involving several invariants, at least one of which is defined homolog-ically. Equation (HI1) is of course an example, and we note that other homologi-cal identities containing the Nakayama automorphism with interesting applicationshave appeared in [BZ, Theorem 0.3] and [CWZ, Theorem 0.1].

Next, we prove a homological identity which shows how the Nakayama automor-phism changes under a graded twist in the sense of [Zh]. Let σ be a graded algebraautomorphism of A and let Aσ denote the left graded twist of A associated to thetwisting system σ := {σn | n ∈ Z}. Recall that Aσ is an algebra with the sameunderlying graded vector space as A, but with new product a∗ b = σ||b||(a)b for ho-mogeneous elements a, b, where || b || indicates the degree of the element b. Then itis easy to check using the properties of graded twists that A is skew CY if and onlyif Aσ is (see Theorem 5.4 below). For a nonzero scalar c, define a graded algebraautomorphism ξc of A by ξc(a) = c||a||a for all homogeneous elements a ∈ A.


Theorem 0.3. Let A be a noetherian connected graded skew CY algebra, and let lbe the AS index of A. Then

(HI2) µAσ = µA ◦ σl ◦ ξ−1hdet(σ).

Note here that hdet is defined using the natural action of the group of gradedautomorphisms H = AutZ(A) on A. Theorem 0.3 will be proved in Section 5. Theresult has many applications; for example, if one wants to prove a result aboutthe Nakayama automorphism, one can often perform a graded twist to reduce tothe case of an algebra with a simpler Nakayama automorphism (for example, seeLemma 6.2 below). Similarly, in some cases one can twist a skew CY algebra toobtain a CY one, and so this gives a method of producing more CY algebras; seeSection 7 for examples.

Another goal of this paper is to demonstrate a strong connection between theNakayama automorphism µA and the homological determinant hdet. This is alreadyevidenced by identities (HI1), (HI2) and [CWZ, Theorem 0.1], and it is reinforcedby the next result.

Theorem 0.4. Let A be a noetherian connected graded Koszul skew CY algebra.Then

(HI3) hdet(µA) = 1.

In fact, in all examples of graded skew CY algebras we have observed, hdet(µA) =1, and we conjecture that this is true in much wider generality. In particular,one may consider AS Gorenstein rings, which satisfy (E0.1.1) but for which thehomological smoothness condition is replaced by the weaker assumption that A hasfinite injective dimension. In fact, homological identities (HI1) and (HI2) are truein the AS Gorenstein case; we give the more general versions in the body of thepaper. We conjecture that (HI3) also holds for all connected graded AS Gorensteinrings (Conjecture 6.4).

The above homological identities have some immediate consequences. The fol-lowing result may be useful in further study of the homological determinant.

Corollary 0.5. Let A be a connected graded skew CY algebra, and assume thatConjecture 6.4 holds. For every ϕ ∈ AutZ(A), then

hdetϕ = (µ(A[t;ϕ])(t)) t−1,

where A[t;ϕ] is the corresponding skew polynomial ring.

The next result provides several methods for constructing CY algebras startingwith skew CY algebras.

Corollary 0.6. Let A be connected graded skew CY and let 〈µA〉 be the subgroupof AutZ(A) generated by µA. Assume that hdetµA = 1.

(a) The skew polynomial ring A[t;µA] is CY.(b) The skew group algebra Ao 〈µA〉 is CY.(c) Suppose that µA has finite order. Then the fixed subring A〈µA〉 is AS Goren-

stein with trivial Nakayama automorphism.

Using homological identity (HI2), one may study the Nakayama automorphismsof the entire family of graded twists of a given skew CY algebra A. Let o(µA) bethe order of the Nakayama automorphism µA. In most cases, one can twist A toproduce an algebra with Nakayama automorphism of finite order.


Corollary 0.7. Assume that the base field k is algebraically closed of characteristiczero. Let A be connected graded skew CY with hdetµA = 1. Then there is aσ ∈ AutZ(A) such that o(µAσ ) <∞.

The paper is organized as follows. Section 1 discusses various sources of examplesof skew CY algebras. Section 2 contains some general results on Hopf algebraactions on algebras and bimodules, as well as smash products. Section 3 includes astudy of the local cohomology of a smash product involving a finitely graded algebra,as well as a discussion of the homological determinant for Hopf actions on certaingeneralized AS Gorenstein algebras. Sections 4–6 respectively contain the proofsof the homological identities (HI1)–(HI3). Section 7 is devoted to applications ofthese homological identities and includes proofs of Corollaries 0.5–0.7. In addition,some questions and conjectures are presented in Sections 4, 6, and 7.

We plan to study more results about the order of the Nakayama automorphismin a second paper [RRZ]. In particular, there is an example of a skew CY algebraA such that Aσ is not CY for any graded algebra automorphism σ [RRZ]. Thus

min{o(µAσ ) | σ ∈ AutZ(A)}is a non-trivial numerical invariant of A. We hope this invariant might be helpful inthe project of classifying Artin-Schelter (AS) regular algebras of global dimensionfour. For example, if A is AS regular of dimension 4 and generated by 2 elementsin degree 1, then, up to a graded twist, we can show that µA is of the form ξc wherec7 = 1 [RRZ].

To close, we note that it would be useful to develop a more general theoryof skew CY algebras parallel to that of CY algebras. For example, much workon CY algebras has focused on showing how these arise in many important casesfrom potentials. While in this paper we do not pursue this point of view, we notethat the theory of potentials has been generalized to the skew CY setting; forexample, “twisted superpotentials” are discussed in [BSW, Section 2.2]. It wouldbe interesting to work out some basic properties of such generalized potentialsand relate them to our results in this paper. We also mention that skew Calabi-Yau categories have been considered by Keller, who suggests that “conservativefunctors” may serve as a good replacement for the identity functor in the definitionof Calabi-Yau categories.

Acknowledgments. The authors thank Raf Bocklandt, Bernhard Keller, TravisSchedler, Paul Smith, Michael Wemyss, and Quanshui Wu for useful conversations.We thank Bernhard Keller for sharing his ideas on defining skew Calabi-Yau cat-egories. We are also grateful to Chelsea Walton and Amnon Yekutieli for helpfulcomments on a draft of this paper. Reyes was partially supported by a Universityof California President’s Postdoctoral Fellowship during this project. Rogalski waspartially supported by the US National Science Foundation grants DMS-0900981and DMS-1201572. Zhang was partially supported by the US National ScienceFoundation grant DMS-0855743.

1. Examples of skew CY algebras

In this section, we set the scene by pointing out a number of known examplesof skew CY algebras. We also review some definitions and terminology which wewill need in the sequel. The skew CY property for noncommutative algebras hasbeen studied under other names for many years, even before the definitions of CY


algebras and categories. If A is Frobenius (finite dimensional, but not necessarilyhomologically smooth), then the automorphism µA as defined in (E0.1.1) is theNakayama automorphism of A in the classical sense. Similar ideas were consideredin [VdB1, YeZ1] for graded or filtered algebras when rigid dualizing complexes werestudied in the late 1990s. The skew CY property was called “rigid Gorenstein” in[BZ, Definition 4.4], was called “twisted Calabi-Yau” in [BSW] (the word “twisted”was also used in the work [DV]), and mentioned in talks by several other people withpossibly other names during the last few years. As mentioned in the introduction,we prefer the term “skew Calabi-Yau”, since we will use the word “twisted” alreadyin our study of graded twists of algebras.

The first major examples of skew CY algebras are the Artin-Schelter regularalgebras. The following definition was introduced by Artin and Schelter in [ASc].

Definition 1.1. A connected graded algebra A is called Artin-Schelter Gorenstein(or AS Gorenstein, for short) if the following conditions hold:

(a) A has finite injective dimension d <∞ on both sides,(b) ExtiA(k,A) = ExtiAop(k,A) = 0 for all i 6= d where k = A/A≥1, and

(c) ExtdA(k,A) ∼= k(l) and ExtdAop(k,A) ∼= k(l) for some integer l.

The integer l is called the AS index. If moreover

(d) A has (graded) finite global dimension d,

then A is called Artin-Schelter regular (or AS regular, for short).

In the original definition of Artin and Schelter, A is required to have finiteGelfand-Kirillov dimension, but this condition is sometimes omitted. It is useful tonot require it here, since CY algebras of infinite GK-dimension are also of interestto those working on the subject. It is known that if A is AS regular, then the ASindex of A is positive [SteZh, Proposition 3.1]. If A is only AS Gorenstein, the ASindex of A could be zero or negative.

It is important to point out that the skew CY and AS regular properties coincidefor connected graded algebras. The following lemma is well-known (at least in thecase when A is noetherian).

Lemma 1.2. Let A be a connected graded algebra. Then A is skew CY (in thegraded sense) if and only if it is AS regular.

Proof. If A is AS regular, then A is homologically smooth by [YeZ2, Propositions4.4(c) and 4.5(a)] and satisfies (E1.0.1) by [YeZ2, Proposition 4.5(b)]. Thus A isskew CY.

Conversely, we assume that A is skew CY. By [YeZ2, Lemma 4.3(b)], the (leftand right) global dimension of A is equal to the projective dimension of AeA, whichis finite by the homological smoothness of A. We now work in the bounded derivedcategory of left Ae-modules. Let U = ExtdAe(A,A

e) = 1Aµ(l) as in (E1.0.1). SinceA is homologically smooth, by [VdB2, Theorem 1],

(E1.2.1) RHomAe(A,N) ∼= U [−d]⊗LAe N

for any left Ae-module N , where [n] means complex shift (or suspension) by degreen. Let σ be any graded automorphism of A and let P • be a graded projectiveresolution of 1Aσ as a left Ae-module. Let k = A/A≥1; since A and k are naturally(A,A)-bimodules, A ⊗ k is an (Ae, Ae)-bimodule, with outer left Ae-action and


inner right Ae-action. Then as elements of the derived category, we have

(A⊗ k)⊗LAe 1Aσ = (A⊗ k)⊗Ae P • ∼= A⊗A P • ⊗A k ∼= P • ⊗A k ∼= 1Aσ ⊗A k ∼= k.

Similarly, in the derived category we have

1Aσ ⊗LAe (A⊗ k) ∼= k.

Now, using the above equations, hom-tensor adjunction and (E1.2.1), we have

RHomA(k,A) ∼= RHomA((A⊗ k)⊗LAe A,A) ∼= RHomAe(A,RHomA((A⊗ k), A))

∼= RHomAe(A,A⊗ k) ∼= U [−d]⊗LAe (A⊗ k)

∼= (1Aµ(l)[−d]) ⊗LAe (A⊗ k) ∼= k(l)[−d]

which means that ExtiA(k,A) =

{0 if i 6= d

k(l) if i = d. By symmetry, a similar statement

is true for ExtiAop(k,A). Therefore A is AS regular. �

By the previous result, the theory of skew CY algebras encompasses all of thetheory of AS regular algebras (even those of infinite GK-dimension). In fact, it ismuch more general still, as it applies to many other types of algebras, includingungraded ones. For example, one may consider Hopf algebras which are skew CY,and we discuss this case next. We first review a few facts about Hopf algebras;some more review will be found in the next section. We recommend [Mo] as a basicreference for the theory of Hopf algebras.

Throughout this paperH will stand for a Hopf algebra (H,m, u,∆, ε) over k, withbijective antipode S. This assumption on S is automatic if H is finite dimensionalover k, or if H is a group algebra kG, and these are two important cases of interest.We use Sweedler notation, so for example we write ∆(h) =

∑h1⊗h2, and further we

often even omit the∑

from expressions. For an algebra homomorphism f : H → k,the right winding automorphism of H associated to f is defined to be

Ξrf : h→∑

h1f(h2)

for all h ∈ H. The left winding automorphism Ξlf of H associated to f is defined

similarly, and it is well-known that both Ξlf and Ξrf are algebra automorphisms

of H. Recall that the k-linear dual H∗ = Homk(H, k) is an algebra with productgiven by the convolution f ? g, where f ? g(h) =

∑f(h1)g(h2). Let f : H → k

be an algebra map. Then the functions f : H → k and f ◦ S : H → k are easilyproved to be inverses in the algebra H∗, and using this, one sees that Ξrf and Ξrf◦Sare inverse automorphisms of H. This also implies that f ◦ S and f ◦ S2 are alsoinverses in H∗, and thus

(E1.2.2) f ◦ S2 = f.

As another consequence, it is easy to check that any winding automorphism com-mutes with S2.

A standard tool in the theory of the finite-dimensional Hopf algebras H is thenotion of integrals [Mo, Chapter 2]. The theory of integrals has been extendedto infinite-dimensional AS-Gorenstein Hopf algebras H in [LuWZ2]. The left ho-mological integral of such an H is defined to be the 1-dimensional H-bimodule∫ l

= ExtdH(Hk,HH), where d = injdim(H). Picking 0 6= u ∈∫ l

, then u is an in-variant for the left H-action, that is, hu = ε(h)u for all h ∈ H, but the right action


gives an algebra map η : H → k defined by u · h = η(h)u which may be nontrivial.

By an abuse of notation we write∫ l

= η, so that one has the corresponding winding

automorphisms Ξr∫ l and Ξl∫ l . One may define the right integral∫ r

= ExtdH(kH , HH)

analogously; the left action on it gives an algebra map∫ r

: H → k which is known

to satisfy∫ r

=∫ l ◦S. Thus Ξr∫ r and Ξr∫ l are inverses in the group of algebra

automorphisms of H, and similarly for the left winding automorphisms.

Lemma 1.3. Let H be a noetherian Hopf algebra. Then H is AS regular in thesense of [BZ, Definition 1.2] if and only if H is skew CY. A Nakayama automor-phism of such a Hopf algebra H is given by S−2 ◦ Ξr∫ l . (Alternatively, S2 ◦ Ξl∫ l is

also a Nakayama automorphism of H.)

Proof. By [BZ, Lemma 5.2(c)], if H is AS regular, then H is homologically smooth.Then H is skew CY, and each of the given formulas is a Nakayama automorphism,by [BZ, Theorems 0.2 and 0.3] and the discussion in [BZ, Section 4]. Conversely,by [LuWZ1, Theorem 2.3], a noetherian skew CY Hopf algebra is AS regular. �

The same argument as in the second half of the proof of Lemma 1.2 shows that anon-noetherian skew CY Hopf algebra is AS regular. It is plausible that the converseis true, but this is unknown. Brown has conjectured that every noetherian Hopfalgebra is AS Gorenstein [Br, Question E]. If Brown’s conjecture holds, then Lemma1.3 (together with [LuWZ1, Theorem 2.3]) implies that the class of noetherianHopf algebras with finite global dimension is equal to the class of noetherian skewCY Hopf algebras. In any case, Hopf algebras clearly give another large class ofexamples of skew CY algebras.

An interesting special case of both of the classes of examples introduced so far isthe class of enveloping algebras of graded Lie algebras. Note that there are manyexamples of graded Lie algebras, for example the Heisenberg algebras.

Example 1.4. Let g be a finite dimensional positively graded Lie algebra. Thenthe universal enveloping algebra U(g) is noetherian, connected graded, AS regular,and CY, of global dimension d = dimk g.

Proof. This is quite well-known. For instance, the CY property of U(g) followsfrom [Ye, Theorem A]. We give a short proof based on Lemma 1.3.

Write U = U(g). Since g is positively graded, the Hopf algebra U is connectedgraded. Since dimk g = d <∞, the algebra U is noetherian. It is well-known thatU is in addition an AS regular, Auslander regular and Cohen-Macaulay domain ofglobal dimension d. See, for example, [FV, Section 2]. Now by Lemma 1.3, thealgebra U is skew CY. Since U is connected graded and the maximal graded idealof U is m := ker ε, we have ExtdU (k, U) ∼= k = U/m as U -bimodules. This impliesthat the left integral of U is trivial and hence Ξr∫ l is the identity map. Since U

is cocommutative, S2 is the identity map. By Lemma 1.3, µU is the identity andhence U is CY. �

The Nakayama automorphism of a skew CY algebra is not in general easy tocompute. For a Frobenius algebra, it is the same as the classical Nakayama auto-morphism and is related to the corresponding bilinear form (see the end of Section3). We have seen that there is a formula for the Nakayama automorphism in thecase of Hopf algebras (Lemma 1.3) which reduces the problem to calculating the


homological integral. If the algebra A is a connected graded Koszul algebra, thenthe Nakayama automorphism of A can be determined if one knows the Nakayamaautomorphism of the Koszul dual A!, which is Frobenius, since the actions of theNakayama automorphisms on the degree 1 pieces of A and A! are dual up to sign[VdB1, Theorem 9.2]. This also works in the more general case of N -Koszul algebras[BM, Theorem 6.3].

The aim of the rest of the paper is to give formulas which help to compute theNakayama automorphism by showing how the Nakayama automorphism changesunder some common constructions. We give one simple result of this type rightnow. Recall that an element z ∈ A is called normal if (i) there is a τ ∈ Aut(A)(the group of algebra automorphisms of A) such that za = τ(a)z for all a ∈ A, and(ii) τ(z) = z. It is clear that, if z is a nonzerodivisor, then (ii) follows from (i).Let σ be in Aut(A). A normal element z is called σ-normal if it is a σ-eigenvector,namely, σ(z) = cz for some c ∈ k× := k \ {0}.

There is a strong connection between the Nakayama automorphism and the the-ory of dualizing complexes, which we need in the next proof and a few others,especially in Lemma 3.5 below. These instances are few enough that we do not re-view the theory of dualizing complexes here; the reader can find the basic definitionsand results in the papers we reference.

Lemma 1.5. Let A be noetherian connected graded AS Gorenstein algebra andlet z be a homogeneous µA-normal nonzerodivisor of positive degree. Let τ be theelement in AutZ(A) such that za = τ(a)z for all a ∈ A. Then µA/(z) is equal toµA ◦ τ restricted to A/(z).

Proof. Let d = injdimA. Considering A as an ungraded algebra, 1Aν [d] is therigid dualizing complex over A, where ν−1 is the Nakayama automorphism of A,by [YeZ1, Proposition 6.18(2)] (see also Lemma 3.5 below and [YeZ2, Proposition4.5(b,c)]). The rigid dualizing complex overA/(z) is equal to RHomA(A/(z), 1Aν [d])by [YeZ1, Theorem 3.2 and Proposition 3.9]. A computation shows that

Ext1A(A/(z), 1Aν) = Ext1A(A/(z), A)ν = {1(A/(z))τ−1

}ν = (A/(z))τ−1◦ν ,

as left Ae-modules, and that Exti(A/(z), A) = 0 for i 6= 1. Thus

RHomA(A/(z), 1Aν [d]) = Ext1A(A/(z), 1Aν)[d− 1] = (A/(z))τ−1◦ν [d− 1].

Thus the Nakayama automorphism of A/(z) is µA ◦ τ restricted to A/(z). �

Recall from the introduction that when A is a Z-graded algebra, for any 0 6=c ∈ k we have the graded automorphism ξc : A → A with ξc(a) = c||a||a for allhomogeneous elements a, where || || denotes the degree of homogeneous elements.A Z-graded skew CY algebra with µA = ξc is called c-Nakayama. In a sense, gradedc-Nakayama algebras are closer to CY algebras than more general skew CY algebras.We now give an explicit example which was one of the motivating examples for ourwork in this paper calculating the Nakayama automorphism of a smash product.It is a simple example of a ring A which is skew CY only, but which becomes CYafter passing to a skew group algebra.

Example 1.6. Let A be the k-algebra generated by x and y of degree 1 and subjectto the relations

x2y = yx2, y2x = xy2.


This is the down-up algebra A(0, 1, 0) [KK, p. 465], which is AS regular algebra oftype S1, with AS index 4 [ASc]. Let z = xy − yx, then one checks that xz = −zxand yz = −zy. Hence z is normal and zf = ξ−1(f)z for all f ∈ A.

Let B = A/(z). Then µB = IdB , as B ∼= k[x, y] is commutative. Since A isZ2-graded with deg(x) = (1, 0) and deg(y) = (0, 1), µA must be Z2-graded. HenceµA sends x to ax and y to by for some a, b ∈ k×. Consequently, µA(z) = λz forsome λ ∈ k×. Applying Lemma 1.5 (viewing A as N-graded), µB = (ξ−1 ◦ µA) |B .Thus µA = ξ−1 when restricted to A/(z). Since both A and A/(z) are generatedby x and y, it follows that µA = ξ−1, or in other words A is (−1)-Nakayama (hencenot CY).

Consider the group G = {1, ξ−1} acting on A naturally. We claim that forH = kG, the smash product A#H (or equivalently, the skew group algebra AoG)is CY. For those familiar with theory of superpotentials, one way to see this is toshow that A o G is isomorphic to a factor algebra of the path algebra kQ, whereQ is the McKay quiver corresponding to the action of G on A1; see [BSW, Section3]. In the case at hand, Q is the quiver given in [Boc, Section 5.3] with vertices{1, 2}, two arrows a1, a2 from 1 to 2, and two arrows a3, a4 from 2 to 1. The algebraAoG is the factor algebra of kQ given by the cubic relations given by taking cyclicpartial derivatives of the superpotential W = a1a3a2a4 + a3a1a4a2. This algebra isshown to be CY of dimension 3 in [Boc, Theorem 5.4].

We have not given full details of the argument above, because one of the goalsof our paper was to seek a deeper reason why a skew group algebra of a non CYalgebra might become CY. This is achieved by Corollary 0.6(b), which shows thatA o G must be CY since G is the cyclic subgroup generated by the Nakayamaautomorphism.

The current example is also interesting in the context of our main theorem onthe Nakayama automorphisms of graded twists. Let σ ∈ AutZ(A) be defined byσ(x) = x and σ(y) = iy where i2 = −1. By [KK, Theorem 1.5], hdetσ = i2 = −1.Since σ4 = 1, by Theorem 0.3 the graded twist Aσ is CY. One may check thatthe algebra Aσ is generated by x and y subject to the relations x2y = −yx2 andy2x = −xy2. One may also use Lemma 1.5 together with a direct computation toshow that Aσ is CY.

If A is a Zw-graded algebra, we will let AutZw(A) denote the group of allgraded algebra automorphisms of A. Starting from Section 2, we will considernon-connected graded skew CY algebras. In this case the Nakayama automor-phism may not be unique, as we see in the following standard lemma whose proofwe leave to the reader.

Lemma 1.7. Let A be a Zw-graded algebra and M be a graded A-bimodule. Supposethat every homogeneous left-invertible (respectively, right-invertible) element of Ais invertible. Let ν, σ denote elements in AutZw(A).

(a) M is isomorphic to 1Aν , for some ν, if and only if MA∼= AA and AM ∼= AA.

(b) Suppose M ∼= 1Aν . Then an element e ∈ M is a generator for AM if andonly if it is a generator for MA.

(c) Let e be a generator of 1Aν . Replacing e to ef for some invertible elementf changes ν to ν ◦ ηf , where ηf is the inner automorphism a 7→ faf−1 forall a ∈ A.

(d) 1Aν is isomorphic to 1Aσ if and only if νσ−1 is an inner automorphism.


Of course in most common situations, for example if A is N-graded and A0 isfinite dimensional, or if A is a domain, then every homogeneous left-invertible(respectively, right-invertible) element is invertible. In the situation of Lemma 1.7,if AM = Ae ∼=A A and MA = eA ∼= AA, then we call e a generator of the A-bimoduleM .

2. Hopf actions on bimodules and smash products

In this section we collect some preliminary material about (bi)-modules oversmash products and dualization. Some of this material is well-known, but we havetried to make our presentation relatively self-contained. A reader who is less familiarwith Hopf algebras may wish to think primarily about the case of group algebraskG on a first reading, since the results of Sections 2-4 are still non-trivial in thatcase.

We maintain the assumptions about Hopf algebras already mentioned in theprevious sections. Let A be a left H-module algebra; by definition [Mo, Definition4.1.1], this means that A is a left H-module such that

h(ab) =∑

h1(a) h2(b) and h(1A) = ε(h)1A

for all h ∈ H, and all a, b ∈ A. Following [Mo, Definition 4.1.3], the left smashproduct algebra A#H is defined as follows. As a k-space, A#H = A⊗H, and themultiplication of A#H is given by

(a#g)(b#h) =∑

a g1(b)#g2h

for all g, h ∈ H and a, b ∈ A. We identify H with a subalgebra of A#H via themap iH : h → 1#h for all h ∈ H, and identify A with a subalgebra of A#H viathe map iA : a → a#1 for all a ∈ A. We note the following useful formula whichdetermines how an element of A and an element of H move past each other:

(E2.0.1) a#h = (a#1)(1#h) = (1#h2)(S−1(h1)(a)#1).

Let B = A#H. Identifying H and A with subalgebras of B, any left B-moduleis both a left A and left H-module. It is easy to check that a k-vector space Mwhich is both a left A and left H-module has an induced left B-module structurerestricting to the given ones on both A and H if and only if the condition

h(am) = h1(a)h2(m)

is satisfied for all h ∈ H, a ∈ A,m ∈ M . As a special case, if A is a left H-modulesuch that h(1A) = ε(h)1A, then A is a left H-module algebra if and only if theleft A-module and the left H-module structures on A induce a left A#H-modulestructure on A.

One can also construct a right-handed version of smash product. We say that Ais a right H-module algebra if it is a right H-module satisfying

(ab)h = ah2bh1 and (1A)h = ε(h)1A

for all h ∈ H and a, b ∈ A. While one could instead define a right H-module algebraby the more natural-seeming rule (ab)h = ah1bh2 , the convention we have chosen willallow us to more easily relate left and right smash products below. Note also thatour convention is to write all right H-actions in the exponent to avoid notationalconfusion. If A is a right H-module algebra, then the right smash product H#A isdefined to be the tensor productH⊗A with multiplication (h#a)(g#b) = hg2#ag1b.


(We use the same symbol # for the left and the right smash products.) As on theleft, a k-vector space is a right H#A-module if and only if it is a right H and rightA-module satisfying (ma)h = mh2ah1 for all m ∈M,a ∈ A, and h ∈ H.

Recall that in a Hopf algebra, the antipode always satisfies the formulas

(E2.0.2) (S ⊗ S) ◦∆ = τ ◦∆ ◦ S and ε ◦ S = ε,

where τ : H ⊗H → H ⊗H is the coordinate switch map (g ⊗ h) 7→ (h ⊗ g) [Mo,Proposition 1.5.10]. Suppose that A is a left H-module algebra. We may make itinto a right H-module algebra by defining ah = S−1(h)(a) for all a ∈ A, h ∈ H;this is easy to check using (E2.0.2). We always use this fixed convention for makinga left H-module algebra into a right one, since with it we have the following niceproperty.

Lemma 2.1. Let A be a left H-module algebra, and make it into a right H-modulealgebra via ah = S−1(h)(a). Then there is an algebra isomorphism Ψ : A#H ∼=H#A given by the formula a#h 7→ h2#ah1 , with inverse Ψ−1 having the formulag#b 7→ g1(b)#g2.

Proof. We have

Ψ((a#g)(b#h)) = Ψ(ag1(b)#g2h) = g3h2#(ag1(b))g2h1

= g4h3#(ag3h2bS(g1)g2h1) = g2h3#ag1h2bh1 ,

while

Ψ(a#g)Ψ(b#h) = (g2#ag1)(h2#bh1) = g2h3#ag1h2bh1 .

Thus Ψ is a homomorphism of algebras. A dual proof shows that the map Ψ−1 :g#b 7→ g1(b)#g2 is a homomorphism H#A→ A#H.

Note that both the formulas for Ψ and Ψ−1 act as the identity on the subalgebrasidentified with H and A. Thus it is obvious that ΨΨ−1 = Ψ−1Ψ = 1, since bothcompositions are clearly trivial when restricted to these subalgebras. �

From now on we identify the left and right smash products A#H and H#Avia the isomorphism of the previous lemma. In particular, note that if M is ak-vector space with left A and H-actions, then the condition h(am) = h1(a)h2(m)is equivalent to M being either a left A#H or a left H#A-module which restrictsto the given actions of A and H. Similarly, the condition (ma)h = mh2ah1 isequivalent to being either a right A#H or a right H#A-module.

One of the main purposes of this section is to discuss actions of Hopf algebras onbimodules, where clearly one would like to require the Hopf action to interact withthe bimodule structure in some nice way. The twist by Si we allow in the followingdefinition gives a slightly more general notion than what we have seen discussed inthe literature. We will see below that the extra generality will be useful to describethe dual of a bimodule with a Hopf action (Proposition 2.7).

Definition 2.2. Let A be a left H-module algebra. If M is an A-bimodule with aleft H-action satisfying

(E2.2.1) h(amb) = h1(a)h2(m)Si(h3)(b)

for all h ∈ H, a, b ∈ A, m ∈ M and some fixed even integer i, then M is called anHSi-equivariant A-bimodule. When i = 0, then M is simply called an H-equivariantA-bimodule.


Remark 2.3. Being an H-equivariant A-bimodule is equivalent to being a leftmodule over a certain algebra Ae oH introduced by Kaygun [Kay, Definition 3.1,Lemma 3.3]. As a vector space, Ae oH = A ⊗ A ⊗H, with product given by theformula

(a1 ⊗ a′1 ⊗ h)(a2 ⊗ a′2 ⊗ g) = a1h1(a2)⊗ h3(a′2)a′1 ⊗ h2g,for all a1⊗a′1⊗h, a2⊗a′2⊗g ∈ AeoH. More specifically, if M is an H-equivariantbimodule, one may check that it is an AeoH-module via (a⊗a′⊗h) ·m = ah(m)a′.

Lemma 2.4. Let M be an A-bimodule with a left and right H-action related by

(E2.4.1) mh = S−i−1(h)(m)

for all m ∈ M,h ∈ H, and some even integer i. Then M is an HSi-equivariantA-bimodule if and only if M is a left A#H-module and a right A#H-module underthe given actions of A and H.

Proof. We have already seen that M being a left A#H-module is equivalent tothe condition h(am) = h1(a)h2(m) for all h ∈ H, a ∈ A,m ∈ M , and that beinga right A#H ∼= H#A-module is equivalent to (mb)h = mh2bh1 for all h ∈ H,b ∈ A, m ∈M . If M is an HSi-equivariant A-bimodule, then in particular we haveh(am) = h1(a)h2(m) and

(mb)h = S−i−1(h)(mb) = S−i−1(h)1(m)Si(S−i−1(h)2)(b)

= S−i−1(h2)(m)S−1(h1)(b) using (E2.0.2)

= mh2bh1 ,

so that M is a left and right A#H-module. The converse is similar. �

Given any H-equivariant A-bimodule M , one can define an A#H-bimoduleM#H with left and right A#H actions given by formulas analogous to the multi-plication in A#H; the proof that it is an A#H-bimodule is virtually the same asthe proof that the multiplication of A#H is associative. See, for example, [LiWZ,Lemma 2.5(1)]. We will need a generalization of this construction where we al-low the bimodule to be HSi-equivariant and we smash with a twist of H by anautomorphism.

Lemma 2.5. Let B = A#H for some left H-module algebra A. Let M be anHSi-equivariant A-bimodule, and make M into a right H-module as in (E2.4.1).Suppose that σ : H → H is an algebra automorphism of H.

(a) Choose a left and right generator u of the H-bimodule σH1, such that hu =uσ(h). Let M#σH1 be the tensor product M ⊗ σH1, with left B-action

(a#g)(m#uh) = ag1(m)#g2uh

and right B-action

(m#uh)(b#k) = mh1(b)#uh2k.

If σ satisfies the property

(E2.5.1) ∆ ◦ σ = (Si ⊗ σ) ◦∆,

then M#σH1 is a B-bimodule.


(b) Choose a left and right generator u of the H-bimodule 1Hσ, such thatσ(h)u = uh. Let 1Hσ#M be the tensor product 1Hσ ⊗M , with left B ∼=H#A-action

(g#a)(hu#m) = gh2u#ah1m

and right B-action

(hu#m)(k#b) = huk2#mk1b.

If σ satisfies the property

(E2.5.2) ∆ ◦ σ = (S−i ⊗ σ) ◦∆,

then 1Hσ#M is a B-bimodule.(c) (E2.5.1) holds for any automorphism of the form σ = Si ◦ Ξrγ , where γ :

H → k is an algebra map. Similarly, (E2.5.2) holds for any automorphismof the form S−i ◦ Ξrγ .

Proof. (a) We leave the proof of this part to the reader, since it is similar to but abit simpler than the proof of part (b).

(b) It is straightforward to see that 1Hσ#M is a left and right B-module, andso we check carefully only that 1Hσ#M is a B-bimodule. Note first that

(am)k = S−i−1(k)(am) = S−i−1(k2)(a)S−i−1(k1)(m) = aS−i(k2)mk1 .

Now we have

[(g#a)(hu#m)](k#b) = (gh2u#ah1m)(k#b) = (gh2uk2#(ah1m)k1b)

= gh2σ(k3)u#ah1S−i(k2)mk1b,

while

(g#a)[(hu#m)(k#b)] = (g#a)(huk2#mk1b) = g(hσ(k2))2u#a(hσ(k2))1mk1b

= gh2σ(k2)2u#ah1σ(k2)1mk1b.

Using the hypothesis that ∆ ◦ σ = (S−i ⊗ σ) ◦∆, applying this formula to k2 wehave ∆(σ(k2)) = S−i(k2)⊗σ(k3), which implies that the two expressions above arethe same.

(c) If σ = Si ◦ Ξrγ , then

∆(σ(h)) = ∆(Si(h1)Si(γ(h2))) = ∆(Si(h1)(γ(h2)))

= Si(h1)⊗ Si(h2)γ(h3) = Si(h1)⊗ σ(h2)

= (Si ⊗ σ) ◦∆(h)

for all h ∈ H. The other calculation is analogous. �

The next goal is to discuss the behavior of the constructions above under k-linear duals. First, we recall the structure of the dual H∗ = Homk(H, k) of afinite-dimensional Hopf algebra H.

Lemma 2.6. Let H be a finite-dimensional Hopf algebra. Then as H-bimodules,H∗ ∼= 1Hσ where σ = µ−1H = S2 ◦ Ξr∫ r .


Proof. Let 0 6= u ∈ H∗ be a left integral of the Hopf algebra H∗ [Mo, Definition2.1.1]. Then by definition, in the algebra H∗ we have gu = ε∗(g)u for all g ∈ H∗,where here ε∗ is the counit of H∗, given by ε∗(f) = f(1H). It is easy to check thatthe fact that u is a left integral is equivalent to

(E2.6.1)∑

h1u(h2) = u(h)1H

for all h ∈ H [DNR, Remark 5.1.2]. By [Mo, Theorem 2.1.3(3) and its proof], u isa generator of the left and right H-module H∗, with Hu ∼= H as left H-modulesand uH ∼= H as right H-modules. Therefore, by Lemma 1.7 there is an algebraautomorphism σ of H and an isomorphism φ : H∗ → 1Hσ of H-bimodules withφ(hu) = h; in other words, we have the formula σ(h)u = uh for h ∈ H. The left-handed version of the computation in the proof of [FMS, Lemma 1.5] now showsthat σ = S2◦Ξr∫ r = S2◦Ξr∫ l ◦S (the formula (E2.6.1) is needed in the computation).

Since H is finite-dimensional, it is also Frobenius, and so σ−1 is its Nakayamaautomorphism µH (see the discussion at the end of Section 3). In fact, then theformula for σ also follows from Lemma 1.3, since

(S2 ◦ Ξr∫ l ◦S)−1 = (Ξr∫ l ◦S)−1 ◦ S−2 = Ξr∫ l ◦ S−2 = S−2 ◦ Ξr∫ l(see the discussion of winding automorphisms and integrals in Section 1). �

In our applications of this section below, usually A will be an N-graded algebraA =

⊕i≥0Ai which is locally finite (dimk Ai < ∞ for all i) and which is a left

H-module algebra, where the action of H respects the grading in the sense thateach Ai is a left H-submodule. We then say that A is a graded left H-modulealgebra. More generally, a graded HSi-equivariant A-bimodule will be a Z-gradedA-bimodule M satisfying Definition 2.2, such that the action of H respects thegrading. In the next result, we see why we defined both a left and right-sidedversion of the bimodule smash construction in Lemma 2.5: taking a k-linear dualnaturally changes from one of the versions to the other. When we are workingwith Zw-graded modules M , unless otherwise noted, by M∗ we will always meanthe graded dual, that is M∗ =

⊕λ∈Zw Homk(Mλ, k), which is naturally again a

Zw-graded vector space (where elements of Homk(Mλ, k) have degree −λ). Notethat when M is locally finite, the graded dual M∗ remains locally finite and theusual isomorphism M∗∗ ∼= M holds.

Proposition 2.7. Let A be a locally finite graded left H-module algebra for a Hopfalgebra H, and let B = A#H. Let M be a locally finite HSi-equivariant A-bimodule,which we make into a right H-module as well as in (E2.4.1). Let M∗ be the gradedk-linear dual of M .

(a) M∗ is a graded HS−i−2-equivariant A-bimodule.(b) Let H be finite-dimensional, and assume that i = 0. Then

(M#H)∗ ∼= (H∗)#M∗ ∼= 1Hσ#M∗

as B-bimodules, where σ = S2 ◦ Ξr∫ r . Here, M#H is the B-bimodule con-

structed in Lemma 2.5(a) and 1Hσ#M∗ is the B-bimodule constructed inLemma 2.5(b).

Proof. (a) Write lh|M for the map M → M given by left multiplication by h.Similarly, rh|M means the right multiplication by h. By assumption, the left and


right H-actions on M satisfy

rh|M = lS−i−1(h)|Mfor all h ∈ H. Note that rh|M∗ = (lh|M )∗ and similarly (rh|M )∗ = lh|M∗ . Then

rh|M∗ = (lh|M )∗ = (rSi+1(h)|M )∗ = lSi+1(h)|M∗

for all h ∈ H. Clearly M∗ is still an A-bimodule and a B-module on both sides,and so it is an HS−i−2 -equivariant A-bimodule by Lemma 2.4.

(b) M∗ is an HS−2-equivariant A-bimodule by part (1), and since σ = S2 ◦ Ξr∫ rsatisfies (E2.5.2) by Lemma 2.5(c), 1Hσ#M∗ is a well-defined B-bimodule. SinceM#H is a B-bimodule, (M#H)∗ is also a B-bimodule.

Since H is finite-dimensional, we can identify (M ⊗H)∗ with H∗ ⊗M∗ as a k-vector space. Since we are taking graded duals and M is locally finite, the bilinearform 〈·, ·〉 given by the natural evaluation map (H∗#M∗) ⊗ (M#H) → k is aperfect pairing. We can identify H∗ with 1Hσ as H-bimodules, by Lemma 2.6,where u ∈ H∗, the left integral of H∗, is a right and left generator with uh = σ(h)ufor all h ∈ H. Thus to complete the proof, we need only verify that under theseidentifications, the left and right B-module structures on (M#H)∗ and 1Hσ#M∗

agree. For this it is enough to prove that the left and right A-module and H-modulestructures all agree. Note that it is most natural to think of B as the right smashproduct H#A in the proof below.

Considering the left H-structure, choose hu#φ ∈ 1Hσ#M∗, (v#w) ∈ M#H,and k ∈ H. We have

〈(k(hu#φ), v#w〉 = 〈(hu#φ), (v#w)k〉 = 〈(hu#φ), (v#wk)〉 = hu(wk)φ(v),

while

〈((k#1)(hu#φ), v#w〉 = 〈((khu#φ), v#w〉 = khu(w)φ(v).

These are the same because khu(w) = hu(wk) by the definition of the left H-actionon H∗.

The agreement of the right A-structures is similarly straightforward and we leavethe proof to the reader.

For the right H-structure, we have

〈((hu#φ)k, v#w〉 = 〈((hu#φ), k(v#w)〉 = 〈((hu#φ), (k1(v)#k2w)〉= hu(k2w)φ(k1(v)),

while

〈((hu#φ)(k#1), v#w〉 = 〈((huk2#φk1), (v#w)〉 = huk2(w)φk1(v)

= hu(k2w)φ(k1(v)).

Since (hu⊗φ) = (h#1)(u⊗φ), and we have proved that the left H-actions agreealready, it is enough to show that the left A-actions agree when acting on an elementof the special form (u#φ), because of the formula (1#a)(h#1) = (h2#1)(1#ah1)which holds in the right smash product. In this case we have

〈a(u⊗ φ),v#w〉 = 〈u⊗ φ, (v#w)a〉 = 〈u⊗ φ, vw1(a)#w2〉= φ(vw1(a))u(w2) = φ(vu(w2)w1(a))

= φ(vu(w)a) by (E2.6.1)

= φ(va)u(w),


while

〈(1#a)(u⊗ φ), v#w〉 = 〈(u⊗ aφ), v#w〉 = (aφ)(v)u(w) = φ(va)u(w).

This completes the proof. �

The next lemma concerns automorphisms of A#H that are determined by au-tomorphisms of A and H.

Lemma 2.8. Let A be a left H-module algebra. Suppose that µ ∈ Aut(A) andφ ∈ Aut(H). Define a k-linear map µ#φ : A#H → A#H by

(µ#φ)(a#h) = µ(a)#φ(h)

for all a ∈ A and all h ∈ H, and define φ#µ : H#A→ H#A similarly. Then thefollowing hold.

(a) µ#φ is an algebra automorphism of A#H if and only if φ#µ is an algebraautomorphism of H#A.

(b) µ#φ is an algebra automorphism of A#H if and only if

(E2.8.1) φ(h)1(µ(b))#φ(h)2 = µ(h1(b))#φ(h2)

for all b ∈ A and all h ∈ H.(c) If φ = S2n ◦ Ξlη ◦ Ξrγ for some algebra homomorphisms η, γ : H → k, then

(E2.8.1) holds if and only if

(E2.8.2) (Ξlη ◦ S2n)(h)(µ(b)) = µ(Ξrη(h)(b))

for all h ∈ H and b ∈ A.

Proof. (a) Recall from Lemma 2.1 that the algebra isomorphisms Ψ : A#H →H#A and Ψ−1 : H#A → A#H are defined by Ψ(a#h) = h2#ah1 and byΨ−1(g#b) = g1(b)#g2. Suppose that µ#φ is an automorphism of H#A. ThenΨ−1(µ#φ)Ψ is an automorphism of A#H, and it is easy to see that φ#µ =Ψ−1(µ#φ)Ψ, since this formula holds for elements of A and elements of H. Theconverse is similar.

(b) By definition, for all a#h, b#g ∈ A#H,

(µ#φ)(a#h)(µ#φ)(b#g) = (µ(a)#φ(h))(µ(b)#φ(g))

= µ(a)φ(h)1(µ(b))#φ(h)2φ(g)

and

(µ#φ)((a#h)(b#g)) = (µ#φ)(ah1(b)#h2g)

= µ(ah1(b))#φ(h2g) = µ(a)µ(h1(b))#φ(h2)φ(g).

The assertion follows by comparing these two equations.(c) By definition,

φ(h) = η(h1)S2n(h2)γ(h3)

for all h ∈ H. Since S2n is a Hopf algebra automorphism of H, we have

∆(φ(h)) = η(h1)S2n(h2)⊗ S2n(h3)γ(h4)

and

h1 ⊗ φ(h2) = h1 ⊗ η(h2)S2n(h3)γ(h4) = Ξrη(h1)⊗ S2n(h2)γ(h3).

Then (E2.8.1) in this case is the condition

(Ξlη ◦ S2n)(h1)(µ(b))#S2n(h2)γ(h3) = µ(Ξrη(h1)(b))#S2n(h2)γ(h3)


for all h ∈ H and b ∈ A. Clearly then (E2.8.2) implies (E2.8.1). Conversely, if(E2.8.1) holds, then applying the winding automorphism Ξrγ◦S to both sides of the

previous displayed equation and then applying 1#ε gives (E2.8.2). �

Remark 2.9. Recall from [BZ] that the Nakayama automorphism of a noetherianAS Gorenstein Hopf algebra H has the form φ = S−2◦Ξrγ . Using this automorphismof H in the previous result, (E2.8.2) becomes

S−2(h)(µ(b)) = µ(h(b))

for all b ∈ A and h ∈ H. This formula is related to (E3.10.2) and Lemma 5.3(a)below.

We conclude this section with a result that will allow us to better understandthe bimodule (M#H)∗ constructed in Proposition 2.7, in the special case thatM∗ ∼= µA1 for an automorphism µ.

Lemma 2.10. Let A be a left H-module algebra and let B = A#H. Let µ : A→ Abe an algebra automorphism of A and suppose that the A-bimodule N = µA1 hasa left H-action making it a HSi-equivariant A-bimodule for some even integer i.Make N a right H-module as in (E2.4.1). Suppose further that N has a left andright A-module generator e which is H-stable in the sense that h(e) ⊆ ke for allh ∈ H, and define η : H → k by h(e) = η(h)e. Suppose that σ : H → H is anautomorphism satisfying (E2.5.2), and let u be a left and right generator for 1Hσ,so that uh = σ(h)u.

(a) The element u#e is a left and right B-module generator of 1Hσ#µA1, wherethis is the B-bimodule constructed in Lemma 2.5(b).

(b) We have 1Hσ#µA1 ∼= ρB1 as B-bimodules, where the automorphism ρ ofB has the following formula:

ρ(a#h) = µ(a)#Ξlη ◦ σ−1(h).

Proof. (a) We work with the right smash product H#A in the proof, since theformulas for the B-bimodule structure on 1Hσ#µA1 are given in terms of it.

It is straightforward to check the identity

(u#e)(g#1) = ug2#eg1 = ug2#S−i−1(g1)(e)(E2.10.2)

= ug2#η(S−i−1(g1))e = ug2η(S−i−1(g1))#e

= uΞlη◦S(g)#e using (E1.2.2).

Also, (u#e)(1#a) = (u#ea) and (g#a)(u#e) = (gu#ae), so it follows that u#e is aleft and right B-module generator. The same formulas easily imply that no nonzeroelement of B kills u#e on either side, so 1Hσ#µA1 is a free B-module of rank 1on each side. Then this bimodule is isomorphic to ρB1 for some automorphismρ : B → B, by Lemma 1.7.

(b) Since Ξlη◦S = (Ξlη)−1, we calculate for any h ∈ H that

(h#1)(u#e) = hu#e

= uσ−1(h)#e

= (u#e)(Ξlη ◦ σ−1(h)#1) by (E2.10.2).

This shows that ρ(h#1) = Ξlη ◦ σ−1(h)#1.


For any a ∈ A,

(1#a)(u#e) = u#ae = u#eµ(a) = (u#e)(1#µ(a))

Hence ρ(1#a) = 1#µ(a).Thus we know how ρ acts on elements of A and in H, and so ρ = Ξlη ◦ σ−1#µ

as an automorphism of H#A. By Lemma 2.8(a) and its proof, we see that as anautomorphism of the left smash product A#H, we have the formula ρ = µ#Ξlη◦σ−1as required. �

3. AS Gorenstein algebras and local cohomology

In this section, we introduce the main technical tool of our approach in this paper,which is the local cohomology of graded algebras. We also define a generalizationof the AS Gorenstein condition to not necessarily connected graded algebras, anddiscuss the homological determinant in this setting.

Let A be a locally finite N-graded algebra and mA be the graded ideal A≥1. LetA − GrMod denote the category of Z-graded left A-modules. Similarly, if A andC are graded algebras, then (A,C) − GrMod is the category of Z-graded (A,C)-bimodules. For each n and each graded left A-module M , we define

ΓmA(M) = {x ∈M | A≥nx = 0 for some n ≥ 1} = limn→∞

HomA(A/A≥n,M)

and call this the mA-torsion submodule of M . It is standard that the functorΓmA is a left exact functor from A − GrMod to itself. Since this category hasenough injectives, the right derived functors RiΓmA are defined and called the localcohomology functors. Explicitly, one has RiΓmA(M) = limn→∞ ExtiA(A/A≥n,M).See [AZ] for more background.

We also consider Zw-graded algebras for a positive integer w. In the Zw-graded setting, the degree of any homogeneous element a is denoted by | a |=(n1, · · · , nw) ∈ Zw. Define a new Z-grading by || a ||=

∑ws=1 ns. In most cases,

we will assume that A is noetherian and a locally finite N-graded algebra withrespect to the || ||-grading. But in a few occasions, we consider more generalZw-gradings (for example, in the discussion of Frobenius algebras at the end ofthis section). The local cohomology functors for Zw-graded algebras A are definedusing the || ||-grading and the same torsion functor ΓmA . This is an endofunctorof the category of Zw-graded modules and thus the local cohomology modules arealso in this category. There is a forgetful functor from the category of Zw-gradedA-modules to the category of Z-graded A-modules. It is easy to check that thelocal cohomological functors ΓmA and the derived functors RdΓmA commute withthe forgetful functor, so we use the same notation for local cohomological functorsin either graded category.

In the next two lemmas we consider the local cohomology of (bi)-modules overa smash product.

Lemma 3.1. Let A be a locally finite N-graded left H-module algebra for a finite-dimensional Hopf algebra H, and let B = A#H. Let i ≥ 0 be an integer and let Cbe another graded algebra. Then as endofunctors of (B,C)-GrMod,

RiΓmA(−) ∼= RiΓmB (−).


Proof. Note first that B is a flat right A-module. In fact, it is obvious from thinkingof B as a right smash product H#A, as in Lemma 2.1, that B is a free right A-module. Then every (graded) injective left B-module is a (graded) injective leftA-module; see [La, Lemma 3.5], or see [KKZ, Lemma 5.1] for a graded version.

Given a graded (B,C)-bimodule M , consider a graded injective (B,C)-bimoduleresolution I• of M . In other words, we take an injective resolution of M in thecategory (B ⊗k Cop)−GrMod, where B ⊗k Cop is also a free right B-module andhence the left modules Ii are injective in both B−GrMod and A−GrMod. Thus wecan use I• to calculate either functor. It is easy to check using the formula (E2.0.1)that for any left B-module M , ΓmA(M) is a mB-torsion B-submodule of M . Iteasily follows that for any (B,C)-bimodule M , we have ΓmA(M) = ΓmB (M), andthat these are (B,C)-sub-bimodules of M . This implies that ΓmA(−) = ΓmB (−)as endofunctors of (B,C)-GrMod. The assertion easily follows by applying thesefunctors to I• and taking homology. �

Lemma 3.2. Let A be a locally finite N-graded left H-module algebra, and letB = A#H. Suppose that M is a graded H-equivariant A-bimodule.

(a) For any integer d ≥ 0, RdΓmA(M) has a natural left H-action and is anH-equivariant A-bimodule.

(b) Assume H is finite dimensional. Then as B-bimodules,

RdΓmB (M#H) ∼= RdΓmA(M#H) ∼= RdΓmA(M)#H,

where the bimodule structure of RdΓmA(M)#H is as in Lemma 2.5(a).

Proof. (a) We have seen in Remark 2.3 that to be a H-equivariant A-bimodule isequivalent to being a left module over the algebra R = Ae o H of Kaygun. SoM is a left R-module, and clearly in fact R is a graded algebra (with H in degree0) and M is a graded left R-module. Then ΓmA(M) is a left B-submodule and aright A-submodule of M , by the same proof as in Lemma 3.1, and it follows thatΓmA(M) is an R-submodule of M .

Take a graded injective left R-module resolution I• of M . Since we have in Rthat (1⊗ b⊗ 1)(a⊗ 1⊗h) = (a⊗ b⊗h), for all a, b ∈ A and h ∈ H, it is easy to seethat R is a free right A-module. Thus I• is also a graded injective left A-moduleresolution of M , by the same argument as in Lemma 3.1. Thus we may use thisresolution to calculate RdΓmA(M). We conclude that RdΓmA(M) retains a left R-module structure, in other words, it is a graded H-equivariant A-bimodule. In fact,it is then easy to see that to obtain the left R-module structure on RdΓmA(M), wecan use any resolution of M by a complex of graded R-modules each of which isgraded injective as an A-module (in any words, the modules need not be injectiveas R-modules.

(b) The first isomorphism follows from Lemma 3.1. We note that given an left R-module homomorphism φ : M → N , we can take the smash product of each modulewith H as in Lemma 2.5(a) to obtain a map φ#1 : M#H → N#H. Smashingwith H is easily seen to be functorial in the sense that φ#1 is a B-bimodule map.

Again choosing a graded injective left R-module resolution I• of M , we cansmash this resolution with H; this gives a complex I•#H of B-bimodules whichis a resolution of A#H. Note that I•#H is a complex of graded injective leftA-modules, since each Ii is graded injective over A as in part (a) and Ii#H isisomorphic as a left A-module to a finite direct sum of copies of Ii (recall that His finite dimensional).


Now it is easy to check that for any graded left R-module N , ΓmA(N#H) =ΓmA(N)#H as subspaces of N#H, and by functoriality since ΓmA(N) is an R-submodule of N , it follows that ΓmA(N)#H is a B-sub-bimodule of N#H. Finally,using the injective resolution I•#H to compute the derived functors, it follows thatRdΓmA(M#H) ∼= RdΓmA(M)#H as B-bimodules, for each d. �

We would like to consider a generalization of AS Gorenstein algebras to the non-connected case, but where the algebra is still locally finite. In this paper we proposethe following definition.

Definition 3.3. Let A be a Zw-graded algebra, for some w ≥ 1, such that itis locally finite and N-graded with respect to the || ||-grading. We say A is ageneralized AS Gorenstein algebra if

(a) A has injective dimension d.(b) A is noetherian and satisfies the χ condition (see [AZ, Definition 3.7]), and

the functor ΓmA has finite cohomological dimension.(c) There is an A-bimodule isomorphism RdΓmA(A)∗ ∼= µA1(−l) (where this is

the graded dual), for some l ∈ Zw (called the AS index) and for some gradedalgebra automorphism µ of A (called the Nakayama automorphism).

Remark 3.4. Other notions of generalized AS regular algebras were introducedfor not necessarily connected graded algebras by Martinez-Villa and Solberg in[M-VS] and by Minamoto and Mori in [MM]. Even if A is noetherian of finiteglobal dimension, our definition above is slightly stronger than the one in [M-VS].

One of our main motivations for introducing the definition of generalized ASGorenstein is that when A is a connected graded AS Gorenstein algebra whichis a graded H-module algebra for some finite-dimensional Hopf algebra H, thenA#H will be generalized AS Gorenstein (Theorem 4.1(b)). The hypotheses in ourdefinition of generalized AS Gorenstein are chosen to make sure that the theory ofdualizing complexes will work as usual. We discuss this in the following lemma,which justifies calling µ in Definition 3.3(c) a Nakayama automorphism of A.

Lemma 3.5. Let A be generalized AS Gorenstein, and let µ ∈ AutZ(A) such that

RdΓmA(A)∗ ∼= µA1(− l)

for some l ∈ Z; namely, µ is a Nakayama automorphism in the sense of Definition3.3. Then we have

ExtiAe(A,Ae) ∼=

{0 i 6= d1Aµ(l) i = d

;

namely, µ is a Nakayama automorphism in the sense of Definition 0.1.

Proof. Van den Bergh’s paper [VdB1] works with connected graded algebras only.However, one can check that the results of [VdB1, Sections 3-8] hold with no es-sential change for a locally finite N-graded algebra. This generalization is similarto the semi-local complete case which is worked out explicitly in [WZ].

Now [VdB1, Theorem 6.3] shows that R := RdΓmA(A)∗[d] ∼= µA1(−l)[d] is abalanced dualizing complex for A. By [VdB1, Proposition 8.2], it is also a rigid du-alizing complex, and this implies in particular by [VdB1, Proposition 8.4] thatR−1 = RHomAe(A,A

e), where R−1 is the inverse of R under derived tensor.


Equivalently, since the inverse of the bimodule µA1 is isomorphic to 1Aµ, we haveR−1 ∼= 1Aµ(l)[−d] and thus

ExtiAe(A,Ae) ∼=

{0 i 6= d1Aµ(l) i = d

,

as claimed. �

We note in the following remark that generalized AS Gorenstein algebras satisfygeneralized versions of properties (b,c) of Definition 1.1.

Remark 3.6. Suppose that A is generalized AS Gorenstein of injective dimension dand AS index l. Then we claim that for a finite-dimensional graded left A-module Sconcentrated in degree 0, we have that Exti(S,A) = 0 if i 6= d, and that Extd(S,A)is a finite-dimensional k-space concentrated in graded degree − l.

To see this, note that A satisfies the hypotheses of (the locally finite version of)[VdB1, Theorem 5.1], and so we have the local duality formula

RΓmA(M)∗ = RHomA(M,RΓmA(A)∗)

for any graded left A-module M . Since we have RΓmA(A)∗ ∼= µA1(− l)[d] in thiscase, taking M = S the claim follows as long as RΓmA(S)∗ is finite-dimensionaland concentrated in graded degree 0. But by the locally finite version of [VdB1,Lemma 4.4], since S is finite-dimensional we have RΓmA(S)∗ = S∗.

Clearly, one also has a right module analog of the comments above. Note thatthis implies in particular that a connected graded generalized AS Gorenstein algebrais a (noetherian) AS Gorenstein algebra in the usual sense.

The homological determinant has been an important tool for understanding thetheory of connected graded AS Gorenstein algebras, especially the invariant theoryof group and Hopf algebra actions. Next, we develop a theory of homologicaldeterminant that will apply, in certain cases, to generalized AS Gorenstein algebras.

Definition 3.7. Let A be a Zw-graded generalized AS Gorenstein algebra. LetA be a graded left H-module algebra for some Hopf algebra H. Since A is an H-equivariant A-bimodule, then RdΓmA(A)∗ is a graded HS−2 -equivariant A-bimoduleby Lemma 3.2 and Proposition 2.7. Suppose further that there is a nonzero elemente ∈ RdΓmA(A)∗ of degree l such that

(i) e is a left and right generator of the A-bimodule RdΓmA(A)∗; and(ii) ke is a left H-submodule.

(a) The element e is called an H-stable generator of RdΓmA(A)∗.(b) Under the condition (i) alone, there is a graded algebra automorphism µ

such that ae = eµ(a) for all a ∈ A. We call such a µ the e-Nakayamaautomorphism of A. Note that by Lemma 1.7, any other Nakayama auto-morphism will differ by an inner automorphism.

(c) We define an algebra homomorphism hdet : H → k by

(E3.7.1) hdet(h) e = h(e)

for all h ∈ H and call the map hdet the e-homological determinant of theH-action.

Note that the e-homological determinant only depends on the || ||-grading, and isindependent of possible choices of Zw-gradings that induce the same || ||-grading,


as long as the choice of e is fixed. When A is connected graded AS Gorenstein, thenany bimodule generator e of RdΓmA(A)∗ is contained in the 1-dimensional degree-l piece and so is automatically H-stable since the H-action respects the grading.The e-Nakayama automorphism is the unique choice of Nakayama automorphism inthis case, and the homological determinant is independent of e. On the other hand,there is no obvious reason why an arbitrary generalized AS Gorenstein algebrashould have an H-stable generator. In future work, we hope to generalize some ofour theorems below to work without this assumption.

Remark 3.8. In [KKZ, Definition 3.3], the homological determinant hdet : H → kis defined, given a finite dimensional Hopf algebra H and a connected graded ASGorenstein algebra which is a graded left H-module algebra. In this case, we claimthat the definition of hdet we gave above coincides with the definition in [KKZ].Both definitions depend on first putting a left H-action on RdΓmA(A). We didthis in Lemma 3.2 by taking a graded injective resolution I• of A over the Kaygunalgebra R = AeoH and noting that the terms of this resolution are graded injectiveover A also. In [KKZ], a B = A#H-module graded injective resolution of A is used.Similarly as in the proof of Lemma 3.2, any B-module resolutions in which the termsare graded injective over A must induce the same B-module structures on the localcohomology, so we get the same B-action on RdΓmA(A), and in particular the sameH-action, in either case.

Then (RdΓmA(A))∗ obtains a right H-action. Choosing a generator e of degreel gives a map η′ : H → k given by eh = eη′(h) and the homological determinant isdefined in [KKZ, Definition 3.3] to be η = η′ ◦ S. In this paper, we use that theright H-action of (RdΓmA(A))∗ induces a left H-action satisfying eh = S(h) · e asin (E2.4.1), since (RdΓmA(A))∗ is a HS−2 -equivariant bimodule. Since η ◦ S2 = ηby (E1.2.2), the two definitions agree. Moreover, it has already been commented in[KKZ, Remark 3.4] that the definition of homological determinant in [KKZ] agreesin the case H = kG is a group algebra with the original definition given in [JoZ].

In general it is not easy to actually compute the homological determinant. Wemention a few examples of connected graded AS Gorenstein algebras for which theanswer is known.

Example 3.9. (a) If A is the commutative polynomial ring k[V ] where A1 = Vis finite dimensional, then hdetσ = detσ |V for all σ ∈ AutZ(A) [JoZ, p.322].

(b) If A is a graded down-up algebra (which is a special kind of AS regu-lar algebra of global dimension 3 generated by 2 degree 1 elements), thenhdetσ = (detσ |A1

)2 by a result of Kirkman-Kuzmanovich [KK, Theorem1.5].

(c) Let A be the skew polynomial ring k−1[x, y] and let σ ∈ AutZ(A) map x toy and y to x. Then hdetσ = − detσ |A1= 1.

(d) If A is Zw-graded and σ is an automorphism which acts on each gradedpiece by a scalar, then we calculate hdet(σ) in Lemma 5.3 below.

(e) Let A be noetherian AS Gorenstein and σ ∈ AutZ(A). If z is a normalnonzerodivisor such that σ(z) = λz for some λ ∈ k×, then by [JiZ, Propo-sition 2.4],

(E3.9.1) hdetσ |A= λ hdetσ |A/(z) .


On the other hand, it is unclear how to calculate hdetσ for an automorphism σ ofan arbitrary AS regular algebra.

In the next result, we see that there is a useful restriction on the interactionbetween a Nakayama automorphism µ of a generalized AS Gorenstein algebra anda Hopf action on the algebra.

Lemma 3.10. Let A be a generalized AS Gorenstein algebra which is a gradedleft H-module algebra for some Hopf algebra H. Then RdΓmA(A)∗ ∼= µA1(−l)is naturally a graded HS−2-equivariant A-bimodule by Proposition 2.7. Assumefurther that RdΓmA(A)∗ has an H-stable generator e, let µ be the e-Nakayamaautomorphism, and let hdet be the e-homological determinant.

Then the identity

(E3.10.1) (Ξlhdet ◦ S−2)(h)(µ(a)) = µ(Ξrhdet(h)(a))

holds for all h ∈ H and all a ∈ A. As a consequence, if H is cocommutative or ifhdet = ε, then

(E3.10.2) S−2(h)(µ(a)) = µ(h(a))

for all a ∈ A and h ∈ H.

Proof. Write η = hdet : H → k for convenience and recall that this is an algebramap. Applying h to eµ(a) = ae and using that µA1(−l) is an HS−2-equivariantA-bimodule, we have

η(h1)eS−2(h2)(µ(a)) = h1(a)η(h2)e = eµ(h1(a)η(h2)).

This implies that

(Ξlη ◦ S−2)(h)(µ(a)) = η(h1)S−2(h2)(µ(a)) = µ(h1(a)η(h2)) = µ(Ξrη(h)(a)),

where we use η ◦ S−2 = η by (E1.2.2). This is (E3.10.1).If η = ε then Ξlη = Ξrη = Id, while if H is cocommutative, then Ξlη = Ξrη and

S2 = 1. In either case, (E3.10.2) is equivalent to (E3.10.1). �

The following interesting result is an immediate corollary.

Theorem 3.11. Let A be noetherian connected graded AS Gorenstein. Then µAis in the center of AutZ(A).

Proof. This follows from applying Lemma 3.10 to the action of H = AutZ(A) onA, and noting that the group algebra H is cocommutative. �

For the rest of this section we consider Frobenius algebras. Let E be a finitedimensional Zw-graded algebra. We say E is a Frobenius algebra if there is anondegenerate associative bilinear form 〈−,−〉 : E × E → k, which is graded ofdegree − l ∈ Zw. This is equivalent to the existence of an isomorphism E∗ ∼= E[− l]as graded left (or right) E-modules. As a consequence, the injective dimension ofE is zero. The vector l ∈ Zw is called the AS index of E. There is a classicalNakayama automorphism µ ∈ AutZw(E) such that 〈a, b〉 = 〈µ(b), a〉 for all a, b ∈ E.Let e = 〈1,−〉 = 〈−, 1〉 ∈ E∗. Then e is a generator of E∗ such that E∗ ∼= µE1(− l)as graded E-bimodules. Note that µ and e are dependent on the choices of thebilinear form, so they are not necessarily unique. See [Mu] for further background.

It is easy to see that if E is N-graded with respect to the || ||-grading, then Eis generalized AS Gorenstein, with R0ΓmE (E)∗ = E∗ ∼= µE1(− l). Even if E isnot N-graded, we still have that E∗ ∼= µE1(− l); taking G to be the subgroup of


AutZw(E) generated by µ, then E is naturally a graded left kG-module algebra,and so E∗ is a graded kG-equivariant E-bimodule. Then Definition 3.7 can beinterpreted for e ∈ E∗ and we have the following.

Lemma 3.12. Keep the notation above. Then e is kG-stable and hdetµ = 1, wherehdet is the e-homological determinant.

Proof. For any b ∈ E,

µ(e)(b) = e(µ−1(b)) = 〈1, µ−1(b)〉 = 〈b, 1〉 = e(b)

which means that µ(e) = e. Since G = 〈µ〉, e is G-stable. The assertion follows bythe definition of hdet (E3.7.1). �

4. Proof of identity (HI1)

The aim of this section is to prove homological identity (HI1) by computingthe Nakayama automorphism of the smash product of an AS Gorenstein algebraA with a finite dimensional Hopf algebra H action. This generalizes and partiallyrecovers a result of Le Meur [LM, Theorem 1], who studies the differential gradedcase, and a result of Liu-Wu-Zhu [LiWZ, Theorem 2.12], where A is assumed tobe N -Koszul and H is involutory (although not necessarily finite-dimensional).Other papers where similar problems are considered include [Fa, IR, WZhu]. Ourapproach differs from the previous ones in several ways: we do not assume finiteglobal dimension, but rather the weaker Gorenstein condition, and our methodsemphasize the techniques of local cohomology.

Theorem 4.1. Suppose that A is a generalized AS Gorenstein algebra of injectivedimension d which is a graded left H-module algebra for a finite-dimensional Hopfalgebra H. Let B = A#H. Suppose that there is an H-stable generator e ∈RdΓmA(A)∗. Let DB and DA be the rigid dualizing complexes over B and over A,respectively.

(a) We have

DB [−d] ∼= RdΓmB (B)∗ ∼= H∗#RdΓmA(A)∗ = 1Hσ#DA[−d],

as B-bimodules, where σ = µ−1H = S2 ◦ Ξr∫ r .

(b) B is generalized AS Gorenstein, with Nakayama automorphism

µB = µA#(Ξlhdet ◦ µH),

where hdet is the e-homological determinant.(c) If A is connected graded AS regular and H is semisimple, then B is skew

CY.

Proof. (a) Recall that as mentioned in the proof of Lemma 3.5, the results in[VdB1] hold for not-necessarily connected but locally finite N-graded algebras; inparticular, they hold for the algebra B. Since A is generalized AS Gorenstein, by[VdB1, Theorem 6.3] we have that DA[−d] = RdΓmA(A)∗ ∼= µA1(l), as complexesconcentrated in degree 0, and RiΓmA(A) = 0 for i 6= d.

Now for any i ≥ 0 we have

RiΓmB (B)∗ = RiΓmA(A#H)∗ = (RiΓmA(A)#H)∗


as B-bimodules, where we have used Lemma 3.2, and the fact that A is an H-equivariant A-bimodule. Thus RiΓmB (B) = 0 for i 6= d, and RdΓmB (B)∗ ∼=(RdΓmA(A)#H)∗. Now we use Proposition 2.7 to identify

(RdΓmA(A)#H)∗ ∼= 1Hσ#RdΓmA(A)∗ = 1Hσ#DA[−d]

as B-bimodules, where σ = µ−1H = S2 ◦ Ξr∫ r by Lemma 2.6. Since DA[−d] =

RdΓmA(A)∗ is an HS−2-equivariant A-bimodule, the final term is a well-definedB-bimodule as in Lemma 2.5(b).

Note that since H is finite-dimensional, B is a finitely generated left and rightA-module, so B is noetherian also. The algebra B satisfies the χ condition andhas finite cohomological dimension, since these properties also pass to a finite ringextension [AZ, Theorem 8.3 and Corollary 8.4]. Thus the hypotheses of [VdB1,Theorem 6.3] also hold for B, so the rigid dualizing complex for B exists andequals RΓmB (B)∗. Finally, this means that we have DB [−d] = RdΓmB (B)∗.

(b) By Lemma 2.10, taking u to be a bimodule generator of 1Hσ, then u#e is agenerator of 1Hσ#µA1(− l), and we have

1Hσ#µA1(− l) ∼= ρB1

as B-bimodules, where ρ = µA#(Ξlhdet ◦σ−1). This implies that DB∼= ρB1[d](− l),

and thus µB = ρ has the claimed formula, and we have verified Definition 3.3(c) forB. We already checked in part (a) that B is noetherian, satisfies χ, and has finitecohomological dimension. By definition a dualizing complex has finite injectivedimension in the derived category of left or right modules [VdB1, Definition 6.1].Since the dualizing complex for B is isomorphic to (a shift of) B as a right or leftB-module, it follows that B has finite injective dimension on both sides. Thus allparts of the definition of generalized AS Gorenstein hold for B.

(c) If A is AS regular and H is semisimple, then A is homologically smoothby Lemma 1.2 and H is homologically smooth since it is semisimple. By [LiWZ,Proposition 2.11], B = A#H is homologically smooth. The rest of the definitionof skew CY for B follows from part (b) and Lemma 3.5. �

Theorem 0.2 (namely, (HI1)) follows immediately from the previous result. Asmentioned earlier, one of the motivations behind our study of the result above wasto better understand examples such as Example 1.6, where A is only skew-CY, butsome skew group algebra AoG becomes CY. The following corollary, which givesa special case of Corollary 0.6, explains this phenomenon.

Corollary 4.2. Let A be noetherian connected N-graded AS regular algebra withNakayama automorphism µ. Suppose that µ has finite order. If σ is a gradedalgebra automorphism of A such that σn = µ for some n, and hdetσ = 1, thenA#kG is CY, where G = 〈σ〉 is the subgroup of AutZ(A) generated by σ.

Proof. Note that σ has finite order. Let H = kG, and let B = A#H, which is thesame as the skew group algebra AoG. The algebraB is skew CY by Theorem 4.1(c).By Theorem 4.1(b), µB = µA#(Ξlhdet ◦ µH). Since hdetσ = 1, the homologicaldeterminant hdet : kG→ k is trivial, and as a consequence, Ξlhdet = IdH . Since His semisimple, µH = IdH [Lo, 1.7(a)]. Thus µB = µA#Id = σn#Id, which is innersince it is conjugation in B by 1#σn. So B is CY. �

One interesting open question is whether Theorem 4.1(c) holds in a more generalsetting.


Question 4.3. Let H be a Hopf algebra and let A be a left H-module algebra,neither of which is necessarily graded. If A and H are skew CY, then is A#H skewCY? If so, what is the Nakayama automorphism µA#H , in terms of µA and µH?

5. Proof of identity (HI2)

The goal of this section is to prove homological identity (HI2), and we considera slightly more general setting. Let A be a Zw-graded generalized AS Gorensteinalgebra. Let G be a subgroup of AutZw(A), so that A is a left kG-module algebrawhere σ(a) has its usual meaning for σ ∈ G ⊆ AutZw(A). In this context, a kG-eqivariant A-bimodule is an A-bimodule M with left G-action, denoted by ασ :M →M for each σ ∈ G, such that

(E5.0.1) ασ(amb) = σ(a)ασ(m)σ(b)

for all a, b ∈ A, all m ∈ M and all σ ∈ G. For a fixed σ, any morphism α ofA-bimodules satisfying (E5.0.1) is also called a σ-linear A-bimodule morphism.

We review the definition of graded twists of Zw-graded algebras and Zw-gradedmodules [Zh]. For simplicity, we only consider the graded twists by automorphismsof the algebra. Let σ := {σ1, · · · , σw} ⊂ AutZw(A) be a sequence of commut-ing Zw-graded automorphisms of A. Recall that |m| denotes the Zw-degree of ahomogeneous element m in a Zw-graded module M . Let v be an integral vector(v1, · · · , vw). Write σv = σv11 · · ·σvww . Define the twisting system associated to σ tobe the set

σ = {σv | v ∈ Zw}.A (left) graded twist of A associated to σ is a new graded algebra, denoted by Aσ,such that Aσ = A as a Zw-graded vector space, and where the new multiplication} of Aσ is given by

(E5.0.2) a} b = σ|b|(a)b

for all homogeneous elements a, b ∈ A. We note that the paper [Zh] works primarilywith right graded twists, but left graded twists are more convenient in our setting.Given a left graded A-module N , a left graded twist of N is defined by the sameformula (E5.0.2) for all homogeneous a ∈ A, b ∈ N and denoted by N σ. Then N σ

is naturally a left Aσ-module, and the functor N 7→ N σ gives an equivalence ofgraded module categories A-GrMod ' Aσ-GrMod [Zh, Theorem 3.1].

Next, we define the left twist of a graded kG-equivariant A-bimodule M . Con-tinue to write σ = {σ1, · · · , σw}, and assume now that each σi is in the center ofG. (Since the σi are assumed to pairwise commute, this additional assumption canbe effected if necessary by replacing G with the subgroup generated by the σi.) Ifv is an integral vector (v1, · · · , vw), write αvσ for

∏ws=1 α

vsσs = ασv . The left graded

twist of M associated to σ, denoted by M σ, is defined as follows: as a Zw-gradedk-space, M σ = M , and the left and right Aσ-multiplication is defined by

(E5.0.3) a}m} b = σ|m|+|b|(a)α|b|σ (m)b

for all homogeneous elements a, b ∈ A = Aσ and m ∈ M . It is routine to checkthat M σ is a left kG-equivariant Zw-graded Aσ-bimodule, where g ∈ G acts bythe same map αg : M → M of the underlying k-space. It is also easy to checkthat the bimodule twist is functorial, in the sense that if φ : M → N is a graded


kG-equivariant A-bimodule map, then the same underlying set map gives a kG-equivariant Aσ-bimodule map φ : M σ → N σ.

Lemma 5.1. Assume that A is finitely graded with respect to the || ||-grading. LetM be a left kG-equivariant Zw-graded A-bimodule. For any d ≥ 0, RdΓmAσ

(M σ) ∼=RdΓmA(M)σ as graded left kG-equivariant Aσ-bimodules.

Proof. The case d = 0 is easy. Namely, ΓmAσ(M σ) = ΓmA(M)σ since both can be

identified with the same subset of M σ.Now if R = Ae o kG is the Kaygun algebra of Remark 2.3, we may find an

injective resolution M → I• in the category of Zw-graded left R-modules. As wehave noted in the proof of Lemma 3.2, this is also a graded injective left A-moduleresolution. Now since the bimodule twist is functorial, we can twist the entirecomplex as in (E5.0.3) to get an exact complex M σ → (I•)σ of kG-equivariantgraded Aσ-bimodules, where the maps are the same underlying vector space maps.Since the bimodule twist (E5.0.3) restricts to the usual left twist as left A-modules,(I•)σ is a a graded injective resolution of M σ as left Aσ-modules by [Zh, Theorem3.1], so we can use it to calculate RdΓmAσ

(M σ). Since ΓmAσ((Ii)σ) = ΓmA(Ii)σ for

each i, the result follows. �

Lemma 5.2. Let M be a kG-equivariant graded A-bimodule. Then there is anisomorphism (M σ)∗ ∼= (M∗)σ of left kG-equivariant graded Aσ-bimodules.

Proof. Note that M∗ is naturally a kG-equivariant graded A-bimodule by Propo-sition 2.7 (since S2 = 1), so (M∗)σ is a kG-equivariant graded Aσ-bimodule. Simi-larly, (M σ)∗ is a kG-equivariant graded Aσ-bimodule.

By definition, as graded k-vector spaces,

(M σ)∗ = M∗,

and we identify these below. We calculate the left and right Aσ-action on (M σ)∗.Let x ∈ (M σ)∗, m ∈M and a ∈ Aσ. Let ◦ denote the left and right Aσ-actions on(M σ)∗, and let 〈 , 〉 be the canonical bilinear form M∗ ×M → k. Then

〈x ◦ a,m〉 = 〈x, a}m〉 = 〈x, σ|m|(a)m〉 = 〈xσ|m|(a),m〉

= 〈xσ−|x|−|a|(a),m〉, since both sides are zero if |m|+ |x|+ |a| 6= 0.

Hence x ◦ a = xσ−|x|−|a|(a). For the left action, we have

〈a ◦ x,m〉 = 〈x,m} a〉 = 〈x, α|a|σ (m)a〉 = 〈x, α|a|σ (mσ−|a|(a))〉

= 〈(α∗σ)|a|(x),mσ−|a|(a)〉 = 〈σ−|a|(a)(α∗σ)|a|(x),m〉.The natural kG-action on M∗, denoted by αγ |M∗ for all γ ∈ G, satisfies

αγ |M∗= (αS(γ))∗ = [(αγ)∗]−1

since S(γ) = γ−1. Then

〈σ−|a|(a)(α∗σ)|a|(x),m〉 = 〈σ−|a|(a)(ασ |M∗)−|a|(x),m〉and consequently,

a ◦ x = σ−|a|(a)(ασ |M∗)−|a|(x).

Combining the two calculations above, it follows that the Aσ-bimodule (M σ)∗

has left and right Aσ actions satisfying the rule

(E5.2.1) a ◦ n ◦ b = σ−|a|(a)α−|a|σ (n)σ−|a|−|n|−|b|(b)


for all homogeneous elements a, b ∈ A = Aσ and homogeneous element n ∈ N =

M∗. It is now straightforward to check that the function φ : n→ α−|n|σ (n) satisfies

φ(a } n } b) = a ◦ φ(n) ◦ b for all homogeneous n ∈ N , a, b ∈ A. Thus φ gives anisomorphism from (M∗)σ to (M σ)∗ (identifying the underlying k-space of each withM∗). It is also clear that φ respects the left kG-action, since the σi are in the centerof G. Thus φ is the required isomorphism of kG-equivariant Aσ-bimodules. �

For any δ = (δ1, · · · , δw) ∈ (k×)w and any v = (v1, · · · , vw) ∈ Zw, write δv for∏ws=1 δ

vss . Given a δ ∈ (k×)w, a graded algebra automorphism ξδ of A is defined by

ξδ(a) = δ|a|a

for all homogeneous elements a ∈ A. Note that ξδ is in the center of AutZw(A). Ifδ = (c, c, · · · , c), then ξδ = ξc as defined in the introduction.

Lemma 5.3. Let A be Zw-graded generalized AS Gorenstein and suppose that Gis some subgroup of AutZw(A). Retain the notation as above.

(a) Suppose G consists of automorphisms of the form ξδ. Then every homoge-neous generator e of RdΓmA(A)∗ is kG-stable and the e-homological deter-minant satisfies hdet ξδ = δl.

(b) If RdΓmA(A)∗ has a kG-stable generator e, then every element of G com-mutes with the e-Nakayama automorphism µ (in AutZw(A)).

Proof. (a) Every Zw-graded A-bimodule M has a canonical kG-equivariant bimod-ule structure where ξδ(x) = δ|x|x for all homogeneous x. If we take any gradedinjective resolution of I• of A as left Ae-modules, then making each Ii into a kG-equivariant bimodule in the canonical way, it is easy to see that the morphismsin the complex automatically respect the kG action. Then I• is already a gradedR-module resolution of A, where R = Ae o kG is the Kaygun algebra, and so I• isalso an injective resolution in A−GrMod. As noted in the proof of Lemma 3.2, wecan use I• to calculate the R-module structure on M := RdΓmA(A). Thus we seethat the induced kG-action on M is also the canonical one.

For an element g ∈ G, the left g-action on RdΓmA(A)∗ is induced by the rightg action on RdΓmA(A), which is the same as the left S(g) = g−1-action. Sincealso elements of Homk(Mv, k) have degree −v, we see that the left kG-action onRdΓmA(A)∗ is also the canonical one. Now let e be any homogeneous generator of(RdΓm(A))∗. Then the degree of e is l and the kG action on e is the canonical onegiven by

ξδ · (e) = δle

for each ξδ ∈ G. Thus e is G-stable and hdet ξδ = δl.(b) This is a special case of (E3.10.2). �

We are now ready to prove our main result determining the Nakayama automor-phism of a graded twist.

Theorem 5.4. Let A be Zw-graded generalized AS Gorenstein of AS index l. LetG ⊂ AutZw(A), and assume that e is a kG-stable generator of RdΓmA(A)∗. LethdetA be the e-homological determinant, and let µA be the e-Nakayama automor-phism. Suppose that σ = (σ1, . . . , σw) is a collection of graded automorphisms in thecenter of G, and let hdet(σ) denote the vector (hdet(σ1), · · · ,hdet(σw)) ∈ (k×)w.For convenience of notation write m = mA and mσ = mAσ .


(a) The algebra Aσ is generalized AS Gorenstein. Under the isomorphism

ψ : (RdΓmσ (Aσ))∗ ∼= (RdΓm(A)σ)∗ ∼= ((RdΓm(A))∗)σ

coming from Lemmas 5.1 and 5.2, e′ = ψ−1(e) is a kG-stable generator of(RdΓmσ (Aσ))∗. If µAσ is the e′-Nakayama automorphism, then

µAσ = µA ◦ σl ◦ ξ−1hdet(σ).

(b) Let hdetAσ be the e′-homological determinant. Then hdetA(τ) = hdetAσ (τ)for all τ ∈ G.

(c) If µA ∈ G, then hdetAσ µAσ = hdetA µA.

Proof. (a) Since the isomorphism (RdΓmσ (Aσ)) ∼= (RdΓm(A)σ) in Lemma 5.1 isan isomorphism of kG-equivariant bimodules, taking duals we also get an isomor-phism respecting the kG-action. The isomorphism of Lemma 5.2 also respects thekG-action, so both isomorphisms making up ψ are isomorphisms of graded kG-equivariant bimodules. Thus e′ must be a kG-stable generator of (RdΓmσ (Aσ))∗,of degree l.

By the generalized AS Gorenstein property forA, we haveRdΓm(A)∗ ∼= µAA1(− l),and so µAA1(− l) has a kG-stable generator corresponding to e ∈ RdΓm(A)∗, whichwe also call e. Combining this isomorphism with ψ we have

(RdΓmσ (Aσ))∗ ∼= (µAA1(− l))σ,

where e′ corresponds to the kG-stable generator e of (µAA1(− l))σ.We calculate the left and right action on (µAA1(− l))σ. For every a ∈ Aσ, we

have

a} e = σl(a)e = eµA(σl(a))

and

e} a = α|a|σ (e)a = ασ|a|(e)a = e hdet(σ|a|)a

since by definition the action of ασ on e is given by the homological determinant.Hence (µAA1(− l))σ is isomorphic to φ(Aσ)1 where φ(hdet(σ|a|)a) = µA(σl(a)).Thus

φ = µA ◦ σl ◦ ξ−1hdet(σ).

The formula for µAσ follows. In particular, part (c) of the definition of generalizedAS Gorenstein holds for Aσ, and the other parts of the definition are easy to checkusing the properties of graded twists [Zh].

(b) This follows immediately from the fact that e′ and e correspond under anisomorphism of kG-equivariant bimodules.

(c) Since ξhdet(σ) is in the center of AutZw(A) and e is always ξhdet(σ)-stable, wecan replace G with the group generated by ξhdet(σ) and G without changing thehypotheses. Then by parts (a,b) and the fact that

hdet(σl ◦ ξ−1hdet(σ)) = (hdet∏sσ

lss )(

∏s hdet(σs)

ls)−1 = 1,

the assertion follows. �

The proof of homological identity (HI2) is now an easy special case of the pre-ceding theorem.


Proof of Theorem 0.3. Let G be the subgroup of Aut(A) generated by σ. Then Gis abelian, so in particular σ is in the center of G. Since A is connected graded,any generator e of RdΓmA(A)∗ is G-stable. The assertion follows from Theorem5.4(a). �

We close this section with a simple example.

Example 5.5. Let A := kpij [x1, · · · , xw] be the skew polynomial ring generatedby x1, · · · , xn subject to the relations

xjxi = pijxixj

for all i < j, where {pij}1≤i<j≤w is a set of nonzero scalars. Then one checks imme-diately that A is isomorphic to the Zw-graded twist of the commutative polynomialring B = k[x1, . . . , xw] by σ = (σ1, · · · , σw) where σi is defined by

σi(xs) =

{pisxs i < s

xs i ≥ s, for all s.

It is easy to see that l = (1, 1, . . . , 1), and of course µB = 1. We calculatethat σl(xs) =

∏i σi(xs) =

∏a<s pasxs and using Lemma 5.3(a), we get that

ξhdet(σ)(xs) = hdet(σs)xs =∏b>s psbxs. Combining these calculations, by The-

orem 5.4(a) we have

µA(xs) = (∏a<s

pas∏b>s

p−1sb )xs

for all s = 1, 2, · · · , w.If w = 2, kp12 [x1, x2] is a Z-graded twist of k[x1, x2]. For w ≥ 3, kpij [x1, · · · , xw]

is not in general a Z-graded twist of any commutative ring. In fact, it is easy tocheck that kpij [x1, · · · , xw] is a Z-graded twist of B if and only if there is a set

of nonzero scalars {p1, · · · , pw} such that pij = pip−1j for all i, j. In particular,

this example demonstrates why it is useful to consider the generality of Zw-gradedtwists as we have done in this section.

6. Proof of Identity (HI3)

The goal of this section is to prove homological identity (HI3), which we recallstates that the homological determinant of the Nakayama automorphism is 1. Weonly consider the connected graded case for simplicity, and we prove it only forKoszul AS regular algebras in this paper. We do show, however, how our formulafor the Nakayama automorphism of a graded twist from the previous section allowsone to reduce to the case of an algebra with simple Nakayama automorphism of theform ξc. This same method may be useful to prove the result in general.

We start with a general lemma about tensor products of Gorenstein algebras.Let l(A) denote the AS index of an AS Gorenstein algebra A.

Lemma 6.1. Let A and B be noetherian connected graded AS Gorenstein algebras.Suppose that A⊗B is noetherian.

(a) The algebra A ⊗ B is AS Gorenstein and we have µA⊗B = µA ⊗ µB andl(A ⊗ B) = l(A) + l(B) as N-graded algebras. The algebra A ⊗ B is alsoZ2-graded, where the (i, j) graded piece is Ai ⊗ Bj, and with this grading,l(A⊗B) = (l(A), l(B)).

(b) Let σ ∈ AutZ(A), τ ∈ AutZ(B). Then hdetA⊗B(σ⊗τ) = (hdetA σ)(hdetB τ).


Proof. (a) Let d1 = injdimA and d2 = injdimB. By [VdB1, Theorem 7.1],

(E6.1.1) Rd1+d2ΓmA⊗B (A⊗B) ∼= Rd1ΓmA(A)⊗Rd2ΓmB (B),

and this is a rigid dualizing complex for A⊗B. Since A and B are AS Gorenstein,we have Rd1ΓmA(A)∗ ∼= µAA1(− l(A)) and Rd2ΓmB (B)∗ ∼= µBB1(− l(B)), and thusRd1+d2ΓmA⊗B (A⊗B)∗ ∼= µA⊗µB (A⊗B)1(− l(A)− l(B)). Now we note that A⊗Bsatisfies Definition 3.3, since the χ condition for A ⊗ B is part of the existence ofthe dualizing complex ([VdB1, Theorem 6.3]), and A⊗B must have finite injectivedimension since a dualizing complex always does by definition. Thus A ⊗ B isgeneralized AS Gorenstein and so is also AS Gorenstein in the usual sense byRemark 3.6. We have already calculated its Nakayama automorphism and ASindex above. The proof of the multigraded result is the same, since (E6.1.1) alsoholds as Z2-graded modules.

(b) Let G1 = 〈σ〉 and G2 = 〈τ〉. Then G = G1 × G2 is naturally a subgroupof AutZ(A ⊗ B). Recall that the kG structure on Rd1+d2ΓmA⊗B (A ⊗ B) can becalculated using an (A ⊗ B)#kG-injective resolution of A ⊗ B, as in Remark 3.8.If I• is an A#kG1 graded injective resolution of A and J• is an B#kG2 gradedinjective resolution of B, then the tensor product complex I• ⊗k J• is a resolutionof A ⊗ B in the category of graded (A ⊗ B)#k(G1 × G2)-modules. By [VdB1,Lemma 4.5], we have RΓmA⊗B = RΓmA ◦ RΓmB . Also, since we assume A andB are noetherian, direct sums of injective modules over these rings are injective.Then each term of I• ⊗k J• is an injective B-module. Applying ΓmB the resultingcomplex I• ⊗ΓmB (J•) consists of injective A-modules and so applying ΓmA we getthat ΓmA(I•) ⊗ ΓmB (J•) is equal to RΓmA⊗B (A ⊗ B). Now by construction, theaction of G1 ×G2 on the complex

ΓmA(I•)⊗ ΓmB (J•) ∼= µAA1(− l(A))[−d1]⊗ µBB1(− l(B))[−d2]

simply comes from the obvious action induced by G1 acting on the first tensorcomponent and G2 acting on the second tensor component. Taking k-linear duals,the result follows from the definition of hdet. �

We now show how to pass from an arbitrary AS Gorenstein algebra to a closelyrelated one with simpler Nakayama automorphism.

Lemma 6.2. Let A be a connected graded AS Gorenstein algebra. Possibly afterreplacing the base field k with a finite extension, there is a connected graded ASGorenstein algebra B such that hdetµA = hdetµB, and where µB = ξc for somec ∈ k. Moreover, B is a multigraded twist of a commutative polynomial extensionof A.

Proof. Consider first a noetherian connected N2-graded generalized AS Gorensteinalgebra C with AS index l = (l, 1) 6= (−1, 1), and let µC be the Nakayama auto-morphism of C. Define σ = {Id, µ−1C }. By Theorem 5.4(a), µCσ = ξ−1hdetσ = ξ1,cwhere c = hdetµC . Since l 6= −1, write c = dl+1 for some d ∈ k, which we can doafter replacing k by a finite extension field if necessary. (It is not hard to see thatall properties of the algebras are preserved by a finite extension of the base field.)Since hdet is independent of the grading, we now view D = C σ as a connectedN-graded algebra. Let τ = ξ1,d−1 . Then by Theorem 5.4(a), µDτ = ξ−1hdet τ = ξc′ forsome c′ ∈ k×. Finally, by Theorem 5.4(c), hdetµDτ = hdetµD = hdetµC .


Now suppose that A is a noetherian connected graded AS Gorenstein algebrawith AS index l 6= −1. Let C = A ⊗ k[x], which is a noetherian Z2-graded gener-alized AS Gorenstein algebra with AS index (l, 1) 6= (−1, 1), by Lemma 6.1(a). Bythe previous paragraph, there is an algebra B (which is a sequence of graded twistsof C) such that hdetµB = hdetµA and µB = ξc for some c. Note that B is still ASGorenstein, since polynomial extensions preserve this property by Lemma 6.1(a),and it is standard that graded twists do also.

If instead A is noetherian connected graded AS Gorenstein algebra with AS indexl = −1, then first replace A by A[y] which is connected graded AS Gorenstein withAS index l′ = l+1 6= −1, and has hdetµA[y] = hdetµA by Lemma 6.1(b), sinceobviously µk[y] = 1. Then proceed as in the previous paragraph.

In all cases, we see that B exists as stated, where B is a series of (multi)-gradedtwists of a polynomial extension of A in one or two variables. �

Now we are ready to prove homological identity (HI3). The main idea is thatfor Koszul AS regular algebras A, the Nakayama automorphism of A is relatedin a known way to the Nakayama automorphism of its Ext algebra E, which isFrobenius, and so Lemma 3.12 applies.

Theorem 6.3. Let A be a noetherian connected graded Koszul AS regular algebra.Then hdetµA = 1.

Proof. Let d be the global dimension of A. Let E = E(A) =⊕

i≥0 ExtiA(k, k) be

the Ext algebra of A. Since A is Koszul, we may writeA = T (V )/(R) for a space ofrelations R ⊆ V ⊗2, and E is isomorphic to the Koszul dual A! of A, T (V ∗)/(R⊥),which is a finite-dimensional algebra generated in degree 1. Moreover, since A isAS regular, it is known that the algebra E is a Frobenius k-algebra [Sm]; Let νdenote its (classical) Nakayama automorphism. Then by [VdB1, Theorem 9.2] andLemma 3.5, the AS index of A is l = d and one has

µA|V = ξd+1−1 ◦ (ν|V ∗)∗.

By Lemma 6.2, by extending the base field if necessary and replacing A with amultigraded twist of a polynomial extension of A, we may reduce to the case thatµA = ξc for some c. (Note that all of these changes preserve the AS regular Koszulhypothesis.) Then by the formula above, since µA|V is multiplication by c, we getthat ν|V ∗ is multiplication by (−1)d+1c. Since E is generated in degree 1, ν acts onthe 1-dimensional degree d = l piece by the scalar ((−1)d+1c)d = (−1)d(d+1)cd = cd.By Lemma 3.12, hdet ν = cd = 1. On the other hand, we also have that hdetµA =cl = cd by Lemma 5.3, so hdetµA = 1. �

As already alluded to in the introduction, we conjecture that the following moregeneral result holds.

Conjecture 6.4. Let A be a noetherian connected graded AS Gorenstein algebra.Then hdetµA = 1.

7. Applications

We explore some applications of our results above in this section, includingCorollaries 0.6 and 0.7. We concentrate here on connected Zw-graded AS Goren-stein algebras A. Some of these results depend on knowing that hdetµA = 1, as inConjecture 6.4.


We start with some applications to the calculation of the homological determi-nant. Note that Corollary 0.5 is a special case of part (b) of the next result.

Lemma 7.1. Suppose that Conjecture 6.4 holds. Let A be a noetherian connectedgraded AS Gorenstein algebra.

(a) Suppose that z is a homogeneous µA-normal nonzerodivisor of positive de-gree in A, so that µA(z) = cz for some c ∈ k×, and let τ ∈ AutZ(A) be theautomorphism such that za = τ(a)z for all a ∈ A. Then

hdet τ = c.

In particular, if A has trivial Nakayama automorphism, then hdet τ = 1.(b) Let ϕ ∈ AutZ(A). Then

hdetϕ = µA[t;ϕ](t) t−1.

Proof. (a) First note that τ(z) = z. By Lemma 1.5, µA/(z) = (µA ◦ τ) |A/(z).Since A/(z) is also AS Gorenstein, by assumption we have that hdetA/(z) µA/(z) =hdetA/(z)(µA ◦ τ) = 1. By (E3.9.1),

hdetA(µA ◦ τ) = chdetA/(z)(µA ◦ τ) |A/(z)= c

as (µA ◦ τ)(z) = µA(z) = cz. Since hdetA µA = 1, we have

hdetA τ = hdetA(µA ◦ τ) = c.

The last sentence is a special case.(b) Let B = A[t;ϕ]. Note that B is Z2-graded, with degree (i, j)-piece Ait

j . ThusµB is a Z2-graded automorphism, so µB(t) = ct for some c ∈ k×. Let τ ∈ AutZ(B)be defined by τ(atn) = ϕ(a)tn for all a ∈ A and n ≥ 0. Thus tb = τ(b)t for allb ∈ B. By (E3.9.1),

hdetB τ = 1 hdetA ϕ = hdetA ϕ

since τ(t) = 1t. Now applying part (a), one sees that

hdetA ϕ = hdetB τ = c = µB(t)t−1,

as claimed. �

One of our goals for future work is to explore how one might define homologicaldeterminant for automorphisms of not necessarily connected graded algebras. Thissuggests the following question.

Question 7.2. Do Theorems 0.2 and 0.3 suggest a way to define the homologicaldeterminant hdet in a more general setting? For example, µA#H(µA#µH)−1 shouldbe 1#Ξlhdet. If A is local (not graded), this could be a way of defining hdet.

Another possibility is to use the formula in Proposition 7.5(c),

hdetϕ = µA[t;ϕ](t)t−1.

In the ungraded case, one can show that µA[t;ϕ](t)t−1 ∈ A× for any ϕ ∈ Aut(A).

We next study several constructions which help to produce algebras with a trivialNakayama automorphism, in particular, CY algebras.

Proposition 7.3. Let A be a noetherian connected graded AS regular algebra withhdetµA = 1. Then the algebras B = A[t;µA] and B′ = A[t±1;µA] are CY algebras.


Proof. Since A is AS regular, it is well-known that the Ore extension B = A[t;µA]is also. So A[t;µA] is skew CY by Lemma 1.2. Let C = A[t], which as in the proofof Lemma 7.1(b) is Z2 graded. Let σ = (IdC , µ

−1C ). Then the AS index of C is

l(C) = (l(A), 1) by Lemma 6.1(a), and hdetµC = hdetµA = 1, by Lemma 6.1(b).Now by Theorem 5.4(a),

µCσ = µC ◦ σl ◦ ξ−1hdet(σ) = IdC .

Thus C σ is CY. Finally, one may check that the function ψ : C σ → A[t;µA] givenby the formula

∑i ait

i 7→∑i µ

i(ai)ti is an isomorphism of rings.

(b) The ring B′ is the localization of B at the set of powers of the normalnonzerodivisor t. It is standard that such a localization of a CY algebra remainsCY. �

Proof of Corollary 0.6. (a) This is Proposition 7.3(a).(b) If µA has infinite order, the assertion is equivalent to Proposition 7.3(b). If

µA has finite order, then the assertion follows from Corollary 4.2 by taking n = 1.(c) Since µ = µA has finite order and hdetµ = 1, the invariant subring AG is AS

Gorenstein by [JoZ, Theorem 3.3], where G is the finite group 〈µ〉. Let C = AG

and let∫

= 1|G|

∑g∈G g. By [KKZ, (E3.1.1), Lemmas 3.2 and 3.5(d)],

RdΓmC (C)∗ = RdΓm(A)∗ ·∫

= (RdΓm(A)∗)G = (µA1(−l))G

as C-bimodules, where l = l(A). Note that the restriction of µ to C is the identity.Hence

(µA1(−l))G = 1C1(−l)and therefore the Nakayama automorphism of C is trivial by Lemma 3.5. �

We remark that if A is PI, then Corollary 0.6(a) is a special case of a result ofStafford-Van den Bergh [SVdB, Proposition 3.1].

The results above strongly suggest the following question.

Question 7.4. Do the conclusions of Corollary 0.6 hold for ungraded skew CYalgebras A? In particular, when A is skew CY, is Ao 〈µA〉 always CY?

We have already seen in the proof of Lemma 6.2 how our formula for theNakayama automorphism of a graded twist may allow one to pass to a twist equiv-alent algebra with simpler Nakayama automorphism. We now offer several similarresults about how close one might be able to get to a CY algebra through twisting.

Proposition 7.5. Let A be a noetherian connected graded AS Gorenstein algebra.

(a) Assume k is algebraically closed. Suppose that A is Zw-graded with a setof generators x1, · · · , xw such that deg xi is the ith unit vector in Zw. Ifl(A) 6= 0, then there is an automorphism σ ∈ AutZw(A) such that µAσ = ξcfor some c ∈ k×.

(b) Let ϕ ∈ AutZ(A) and assume that hdetµA = 1. Then there is a Z2-gradedtwist of A[t;ϕ] that has trivial Nakayama automorphism.

Proof. (a) Since A is Zw-graded, its Nakayama automorphism µA is Zw-graded.Thus µA(xi) = aixi for all i = 1, · · · , w. Viewing A as a connected graded algebrawe assume that l := l(A) 6= 0. Since k is algebraically closed, there are δi ∈ k×such that δ− l

i = ai for all i. Then σ− l = µA where σ = ξδ ∈ AutZw(A) ⊂ AutZ(A).The assertion follows from Theorem 5.4(a) (with w = 1).


(b) By Proposition 7.3, A[t;µA] is CY. Since µA is in the center of AutZ(A) byTheorem 3.11, a similar argument as in the proof of Proposition 7.3 shows thatA[t;µA] is a Z2-graded twist of A[t;ϕ]. �

For the rest of this section we prove Corollary 0.7. From now on we assume thatk is algebraically closed of characteristic 0. We use [Bor] as a reference for algebraicgroups. The following two lemmas are presumably known, but we sketch the proofssince we lack a reference.

Lemma 7.6. Let A be a finitely generated Zw-graded algebra, which is locally finiteand connected N-graded with respect to the || ||-grading. Then AutZw(A) is an affinealgebraic group over k.

Proof. Let A be generated as an algebra by homogeneous elements {ai}si=1, whereai ∈ Aαi . Choose a graded presentation F/I ∼= A, where F is a Zw-graded freealgebra with homogeneous generators xi which map to the ai. There is a nat-ural injection i : AutZw(A) →

∏si=1 GL(Aαi), where τ ∈ AutZw(A) corresponds

to the product of the bijections it induces of the graded subspaces containing thegenerating set, since any τ ∈ AutZw(A) is determined by the elements τ(ai). Con-versely, choosing arbitrary bi ∈ Aαi , there is an automorphism τ in AutZw(A) withτ(ai) = bi if and only if for every relation r =

∑dj1,...,jkxj1 . . . xjk ∈ I, we have∑

dj1,...,jkbj1 . . . bjk = 0 in A. It is easy to see that for any relation r this gives aclosed condition on the choice of bi, and intersecting over all relations (or a gen-erating set of the ideal of relations) we get that i is a closed regular embedding.The map i is clearly a group homomorphism, so AutZw(A) is an affine algebraicgroup. �

Lemma 7.7. Let d 6= 0. Let G be an abelian affine algebraic group, and let G0 bethe connected component of the identity.

(a) If G is connected, then for any x ∈ G, there exists y ∈ G such that yd = x.(b) For any x ∈ G, there exists z ∈ G of finite order and y ∈ G0 such that

xz ∈ G0 and xz = yd.

Proof. (a) Since G is abelian, the map φ : G → G defined by φ(y) = yd is ahomomorphism of algebraic groups. The image H is a closed algebraic subgroupof G. Hence the quotient algebraic group P = G/H exists, and this is again affine[Bor, Theorem 6.8]. It is also still abelian and connected since G is. Note thatzd = e for all z ∈ P . We claim that P is the trivial group {e}. Since the elementsof P are all dth roots of e, it follows that P consists of semisimple elements underany embedding of P in a matrix group. Then P can be embedded in a torus (k×)n

for some n, by [Bor, Proposition 8.4]. Since P is connected and a torus containsonly finitely many elements of order d, we must have that P is a point. In otherwords, φ is surjective and the result follows.

(b) We just have to produce an element z of finite order such that xz ∈ G0. Theexistence of y follows from part (a). Let H be the connected component containingx. Choose any w ∈ H and m ≥ 1 such that wm ∈ G0. By part (a), there isv ∈ G0 such that vm = wm. Then z = vw−1 satisfies zm = e, so z has finite order.Moreover, clearly xz ∈ G0, since x ∈ H, w ∈ H, and v ∈ G0. �

Theorem 7.8. Let A be a noetherian connected graded AS Gorenstein algebra withl(A) 6= 0 and with hdetµA = 1. Then there is a σ ∈ AutZ(A) such that µAσ is offinite order.


Proof. By Theorem 3.11, µA is in the center Z of AutZ(A). Note that AutZ(A) isan affine algebraic group by Lemma 7.6 and that Z is a closed algebraic subgroupof AutZ(A). It suffices to work in Z for the rest of the proof.

Let x = µ−1A . By Lemma 7.7(b), there is an z ∈ Z of finite order and y ∈ Z0,such that xz ∈ Z0 and yl = xz where l = l(A). Let σ = y. Then by Theorem5.4(a),

µAσ = µA ◦ σl ◦ ξ−1hdet(σ) = z ◦ ξ−1hdet(σ).

Since yl = xz, we have

hdet(σ)l = hdet(yl) = hdetxhdet z = hdet z

since hdetx = (hdetµA)−1 = 1 by assumption. Since z has finite order, hdet zhas finite order. Thus hdet(σ) has finite order. Finally, we see that µAσ has finiteorder. �

We conclude the paper with the proof of the last corollary from the introduction.

Proof of Corollary 0.7. Since A is AS regular, l(A) > 0 [SteZh, Proposition 3.1].The assertion follows from Theorem 7.8. �

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(Reyes) Bowdoin College, Department of Mathematics, 8600 College Station, Brunswick,ME 04011-8486

E-mail address: [email protected]

(Rogalski) UCSD, Department of Mathematics, 9500 Gilman Dr. #0112, La Jolla,CA 92093-0112, USA.


(Zhang) University of Washington, Department of Mathematics, Box 354350, Seat-

tle, WA 98195-4350, USA.


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